The Global Symmetry Group of Quantum Spectral Beams and ... · The Global Symmetry Group of Quantum Spectral Beams and Geometric Phase Factors ... of quantum mechanical description

Research ArticleThe Global Symmetry Group of Quantum Spectral Beams andGeometric Phase Factors

Elias Zafiris12

1Department of Mathematics University of Athens Panepistimioupolis 15784 Athens Greece2Parmenides Foundation Kirchplatz 1 Pullach 82049 Munich Germany

Correspondence should be addressed to Elias Zafiris ezafirismathuoagr

Received 10 November 2014 Accepted 9 March 2015

Academic Editor Soheil Salahshour

Copyright copy 2015 Elias Zafiris This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a cohomological modelling schema of quantum state spaces and their connectivity structures in relation to theformulation of global geometric phase phenomena In the course of this schema we introduce the notion of Hermitian differentialline sheaves or unitary rays and classify their gauge equivalence classes in terms of a global differential invariant given by the deRham cohomology class of the curvature Furthermore we formulate and interpret physically the curvature recognition integralitytheorem for unitary rays Using this recognition theorem we define the notion of a quantum spectral beam and show that it has anaffine space structure with structure group given by the characters of the fundamental group

1 Introduction

In the foundation of quantum mechanics it is commonlybelieved that phases are not important because a state is notactually described by a vector but by a ray or a projectionoperator so that it can always be removed by a local phase orgauge transformationThis fact constitutes a particular man-ifestation of the gauge principle in quantummechanics and itis intrinsically related to the local abelian phase invariance inthe specification of the state vector of a quantum systemThelocality demand in the formulation of the gauge principle iscrucial because despite the freedom of eliminating the phaselocally by means of a suitable gauge transformation globalphase factors contain gauge invariant physical informationand cannot be removed Moreover due to the Born ruleprobability interpretation of a state vector at a single momentof time physical significance has been assigned only to themodulus or magnitude of a state vector whereas its phasehas been ignored Again although it is true that the notionof phase can be always gauged away locally this is not thecase globally Actually the typical global quantummechanicalobservables are relative phases obtained by interferencephenomena These phenomena involve various splitting andrecombination processes of beams whose global coherenceis measured precisely by some relative phase difference For

example we may think of the simplest case of a beam whichis split into two beams propagating for a period of time andfinally recombined Their interference is always measuredby a global relative phase difference Generally a relativephase can be thought of as the physical attribute measuringthe global coherence between two beams sharing a commoninitial and final point in the space of some control variablesparameterizing the dynamical evolution We note that dueto the functional dependence of the dynamical evolutionon the control variables the state of a quantum systemis parameterized implicitly by some temporal parameterthrough the control variables Consequently it is instructiveto stress the fact that although quantum mechanics maybe locally interpreted in terms of probabilities of events(so that phases do not play any role) globally it is therelative phase differences between beams that have the majorphysical significance Failure to recognize this subtle pointfocusing on the distinctive role of the local and global levelsof quantum mechanical description in relation to physicalobservability and classification of measurable informationcan cause serious interpretational problems

The notion of a global geometric phase factor has beenintroduced in quantum mechanics by Berry [1] and for-mulated in fiber bundle theoretic terms by Simon [2] Therelated notion of a global topological phase factor has been

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 124393 16 pageshttpdxdoiorg1011552015124393

2 Advances in Mathematical Physics

introduced previously by Aharonov and Bohm [3 4] see also[5] The conceptual precursors of this astonishing discoverywhich has been unnoticed in the foundations of quantumtheory for more than 60 years are the work of Pancharatnamin polarization optics and the Aharonov-Bohm effect inelectromagnetism In 1956 Pancharatnam [6] realized thatin order to understand interference phenomena it is notrequired to know the absolute phase but only the relativephase difference between light beams in different states ofpolarization For two light beams this relative phase is givenby the phase argument of their complex-valued scalar innerproduct In 1959 Aharonov and Bohm showed that thereexists a measurable global phase shift in the interferencepattern of two coherent wave functions of a charged particledue to the existence of an electromagnetic field constrainedwithin an infinitely long solenoid even in case that the wavefunctions are localized within regions outside the solenoidwhere the strength of the electromagnetic field vanishes

The general Berry-generation mechanism of an exper-imentally observable global phase factor which is of ageometric or topological origin can be described briefly asfollows A quantum system undergoing a slowly evolving(adiabatic) cyclic evolution retains a trace of its motionafter coming back to its original physical state This trace isexpressed by means of a complex phase factor in the wavefunction (state vector) of the system called Berryrsquos phase orthe geometric phase The cyclic evolution which is better tobe thought of as a periodicity property of the state vector ofa quantum system is driven by a Hamiltonian bearing animplicit time dependence through a set of control variablesFor instance we may think of external electric or magneticfields which define the Hamiltonian parametric dependenceof a charged particle The adiabatic condition defines aconstraint of parallel transport specified by the requirementthat the implicit time dependence of the Hamiltonian issufficiently slow so that the state vector stays in the eigenspaceof the same instantaneous eigenvalue of the HamiltonianIntuitively once the state vector is prepared in an instan-taneous eigenstate of the Hamiltonian with an eigenvaluewhich is separated from the neighboring eigenstates by afinite energy gap then it remains there during its transportwithin a finite temporal period We may think of the spaceof control variables as a slowly varying environment withrespect to which a state vector (eigenvector of the Hamil-tonian localized at the corresponding eigenspace) displays ahistory dependent geometric effect When the environmentreturns to its original state the system also does but foran additional global topological or geometric phase factorDue to the implicit temporal dependence imposed by thetime parameterization of a closed path in the environmentalparameters of the control space this global geometric phasefactor is thought of as a trace of the motion encoding theglobal geometric or topological features of the control spaceThe Berry phase is a complex number of modulus one and isexperimentally observable The two most important featuresregarding the experimental detection of a quantum globalphase are as follows (i) it is a statistical object and (ii) it canbe measured only relatively Thus it becomes observable bycomparing the historical evolution of two distinct statistical

ensembles of systems through their interference pattern TheBerry phase is geometric or topological because it dependssolely on the geometry or topology of the control spacepathway along which the state vector is transported It doesnot depend neither on the temporal metric duration of theevolution nor on the particular dynamics that is applied tothe system

It is instructive to distinguish briefly between geometricand topological global phase factors as follows In the case of asimply connected base space of control variables for examplea sphere the global observable phase factor is of a geometricorigin More precisely it appears due to the curvature of thenonintegrable connection via which the transport of statevectors takes place In the case of a nonsimply connectedspace of control variables a global observable phase factorof a topological origin is always going to appear even if theconnection is integrable The integrability property meansthat the connection does not depend on the closed pathtraced on the control space under the proviso that theseclosed paths belong to the same homotopy equivalence classIntuitively speaking a closed path is homotopically nontrivialif it encloses a hole (eg an inaccessible region) so that itcannot be contracted continuously to a point in the controlspace Depending on the topological connectivity propertiesof this space closed paths can be classified homotopically bymeans of global invariants for example the winding numberof a closed path

The particular significance of the concepts of relativetopological and geometric phase factors from the viewpointof our theoretical scheme is that they mark a distinctivepoint in the history of quantum theory where for the firsttime the significance of global observables as distinct entitiesfrom local observables is realized and made explicit throughprecise physical models which have found concrete experi-mental applications In particular these global observablesfor example global relative phases are obtained via anintegration procedure of some differential gauge potentialover a contour represented as a closed path or loop ona base space of control variables (or localization space)which is implicitly parameterized continuously by an externaltemporal parameter This is usually referred to as a cyclicevolution of a quantum state over a space of parameterswhich is implicitly a continuous or smooth function of time Itis also assumed that this evolution is driven by a Hamiltonianwhich is expressed as a function of the localization spaceparameters bearing an implicit time dependence by consid-ering variation of the temporal parameter Intuitively loopsin the space of control variables have the potential to probethe geometrically or topologically invariant information ofthe localization space for example by encircling a hole or aninaccessible region This nontrivial geometric or topologicalinformation of global significance is measured in terms ofglobal holonomy phase factors via the procedure of liftingclosed paths from the localization base space to the statesor observables defined over it according to some paralleltransport constraint (like the adiabatic one) technicallycalled a connection Due to the implicit time dependenceparameterizing this procedure if we trace continuously a loopon the base space then this loop can be lifted to the implicitly

Advances in Mathematical Physics 3

evolving states represented as sections of a fiber bundleor more generally of a fibration over the base space Theparticular global transformation (group element) undergonefor instance by a state vector when it is parallel-transportedalong a closed curve on the localization space is called theholonomy of the connectionThus in this case the holonomydescribes the global state vector transformation induced bycyclic changes (loops) in the controlling variables

The first fiber bundle theoretic model of Berryrsquos mecha-nism along the previous lines has been formulated by Simon[2] in terms of line bundles Since then almost all of themathematical treatments of geometric phase phenomenathat we are aware of have been formulated exclusively infiber bundle terms see for instance the recent collectivework [7] In this communication we would like to proposea change of perspective from the line bundle models tothe sheaves of their sections There are two fundamentalreasons justifying this move First the physical informationis encoded functionally always in the sections of bundles andthe crucial property is that the sections form sheaves that isimplicate the manner by which local structural informationcan be amalgamated into global ones Second the usualanalytic methods can be effectively formulated in terms ofsheaf cohomology [8 9] which may be conceived as auniversal method of finding global invariants of a physicalcharacter in a homotopy-invariant way More precisely thesheaf cohomology groups measure the global obstructionsfor extending sections from the local to the global level forexample extending local solutions of a differential equationto a global solution The focus of this work is to demonstratethat a sheaf cohomological understanding of the origin ofglobal geometric phase factors is physically significant andin particular paves the way for understanding the globalsymmetry group of a quantum spectral beam For mattersof space we assume some basic familiarity of the readerwith the notion of a sheaf in topology and geometry [8ndash14] The utilization of the methods of sheaf theory andsheaf cohomology to capture the essential aspects of gaugetheories namely electromagnetism and Yang-Mills theorieshas been elaborated in detail by Mallios [15 16] inspiredby the approach suggested by Selesnick [17] We employsimilar sheaf-cohomological concepts targeting the crucialgauge theoretic aspect of quantum mechanics namely thelocal phase invariance of a quantum state vector in relationto the observation of global gauge-invariant topological andgeometric phase factors accompanying the periodic evolutionover a space of control variables

2 Cohomological Description ofQuantum State Spaces

21 Line Sheaves of States From a physical viewpoint theconstruction of a sheaf constitutes the natural outcome of acomplete localization process [18ndash20] see also [21] Generallyspeaking a localization process is being implemented interms of an action of some category of reference or param-eterizing frames on a set containing the information of somephysical attribute This set usually is endowed with some

appropriate algebraic structure for instance we may thinkof a vector space of states or an algebra of observables Forexample we may consider the localization process of the set(Hilbert space basis of vectors) of theHamiltonian eigenstatesof an electron (energy eigenfunctions) with respect to somebase space of control parameters In solid state physics thelocalizing base space is considered to be themomentum space[22] More concretely in solids the energy spectrum of theHamiltonian operator has a band structure meaning that itis piecewise continuousThe energy in each continuous piecedepends on the momentum which varies in the BrillouinzoneThus the Brillouin zone ofmomentumvariables consti-tutes the base localization space of the energy eigenstates ofelectrons This is a continuous topological parameter spacebearing the homotopy of a torus

We consider a pair (119883A) consisting of a paracompact(Hausdorff) topological space of control variables 119883 whichis in general not a simply connected space and a soft sheafof commutative rings A localized over 119883 meaning thatevery section over some closed subset in 119883 can be extendedto a section over 119883 We consider the above pair as theGelfand spectrum [23] of a maximal commutative algebra ofobservablesA which are comeasurablewith theHamiltonianof the quantum system We note that if C is the field ofcomplex numbers then an C-algebraA is a ringA togetherwith a morphism of rings C rarr A (making A into a vectorspace over C) such that the morphism A rarr C is a linearmorphism of vector spaces Notice that the same holds if wesubstitute the field C with any other field for instance thefield of real numbers R We also assume that the stalk A

119909of

germs is a local commutative C-algebra for any point 119909 isin 119883A typical example is the case where 119883 is a smooth manifoldof control variables and A is the C-algebra sheaf of germs ofsmooth functions localized over119883 Togetherwith aC-algebrasheafA we also consider the abelian group sheaf of invertibleelements of A denoted by A

Definition 1 AnA-moduleE is called a locally freeA-moduleof states of finite rank 119898 or simply a vector sheaf of states iffor any point 119909 isin 119883 there exists an open set 119880 of119883 such that

E|119880cong

119898

⨁(A|119880) = (A|

119880)119898

(1)

where (A|119880)119898 denotes the119898-terms direct sum of the sheaf of

C-observable algebrasA restricted to119880 for some119898 isin N Wecall (A|

119880)119898 the local sectional frame or local gauge of states of

E associated via the open coveringU = 119880 of119883

Definition 2 AnA-moduleE is called a locally freeA-moduleof states of rank 1 or simply a line sheaf of states if for anypoint 119909 isin 119883 there exists an open set 119880 of119883 such that

E|119880cong A|119880 (2)

where A|119880

denotes the sheaf of C-observable algebras Arestricted to 119880 We call A|

119880the local sectional frame or local

gauge of states of E associated via the open coveringU = 119880

of119883


Furthermore if for any point 119909 isin 119883 there exists an openset 119880 of119883 such that

E|119880cong C|119880

(3)

and then we call any locally free C-module E of rank 1 acomplex linear local system of rank 1

The notion of a vector sheaf of states generalizes thenotion of a vector space of states when there exists aparametric dependence of the driving Hamiltonian of aquantum system froma topological space of control variablesIt is also clear that the set of sections of a vector sheaf canbe equipped locally with the structure of a Hilbert spaceThe crucial point is that locally every section of a vectorsheaf can be written as a finite linear combination of abasis of sections with coefficients from the local algebra Forexample if 119883 is a smooth manifold of control variables andA is the C-algebra or R-algebra sheaf of germs of smoothfunctions on 119883 then every section can be locally written asa finite linear combination of a basis of sections obtainedfrom the adiabatically parameterized spectral resolution ofthe Hamiltonian with coefficients being real-valued smoothfunctions These functional coefficients stand for the param-eterized eigenvaluesWe note again that the time dependenceis implicitly introduced via time-parameterized paths on 119883It is also the case that the set of sections of every vectorbundle on a topological space (not necessarily a smoothmanifold) forms a vector sheaf of sections localized over thisspace Thus the notion of a vector sheaf of states providesthe most general conceptual and technical framework toconsider in a unifying way all cases where there exists aparametric dependence of some observable (eg the energyrepresented by the Hamiltonian operator) and thus of thestates expressed in the basis of eigenstates of the observableby a space of control variables119883 For example if we considerthe Schrodinger equation

120597120595

120597119905= (120577

119894(119905)) 120595 (4)

where 120577119894isin 119883 at any moment of time and thus at each

point of a time-parameterized path in 119883 we can choose anorthonormal basis diagonalizing the Hamiltonian operator

(120577119894) 120595119899(120577119894) = 119864119899(120577119894) 120595119899(120577119894) (5)

where the 119899th functional coefficient is the 119899th 120577119894-parame-

terized eigenvalue

22 Local Gauge Freedom in terms of Cocycles The mainnovelty of the sheaf theoretic perspective is that it natu-rally leads to an equivalent cohomological understanding ofquantum state spaces which captures the relevant gauge-theoretic aspects precisely by means of cocycles according tothe following

Definition 3 Given a vector sheaf of statesE there is specifieda Cech 1-cocycle with respect to a covering U = 119880 of119883 called a coordinate 1-cocycle in 1198851(UGL(119898A)) (with

values in the sheaf of germs of sections into the groupGL(119898C)) as follows

120578120572 E|119880120572

cong (A|119880120572

)119898

120578120573 E|119880120573

cong (A|119880120573

)119898

(6)

Thus for every 119909 isin 119880120572we have a stalk isomorphism

120578120572(119909) E

119909cong A119898119909

(7)

and similarly for every119909 isin 119880120573 If we consider that119909 isin 119880

120572cap119880120573

then we obtain the isomorphism

119892120572120573(119909) = 120578

120572(119909)minus1

∘ 120578120573(119909) A119898

119909cong A119898119909 (8)

The 119892120572120573(119909) is thought of as an invertible matrix of germs at 119909

Consequently 119892120572120573

is an invertible matrix section in the sheafof germs GL(119898A) (taking values in the general linear groupGL(119898C)) Moreover 119892

120572120573satisfy the cocycle conditions 119892

120572120573∘

119892120573120574= 119892120572120574

on triple intersections whenever they are defined

It is clear in this way that in fiber-bundle theoretic termswe obtain a vector bundle with typical fiber C119898 structuregroup GL(119898C) whose sections form the vector sheaf ofstates we started with In particular for 119898 = 1 we obtain aline bundle 119871 with fiber C structure group GL(1C) cong C(the nonzero complex numbers) whose sections form a linesheaf of states L Clearly by imposing a unitarity conditionthe structure group is reduced to 119880(1) Thus particularly inthe case of a line sheaf of states we have the following

Proposition 4 There exists a bijective correspondencebetween line sheaves of states L and Cech coordinate 1-cocycles119892120572120573

with respect to an open coveringU = 119880 of119883

Llarrrarr (119892120572120573) isin 1198851

(U A) (9)

where 119866119871(1A) cong A is the group sheaf of invertible elementsofA (taking values in C) and 1198851(U A) is the set of coordinate1-cocycles

In physical terminology a coordinate 1-cocycle affects alocal frame (gauge) transformation of a vector sheaf of statesE In particular in the case of a unitary line sheaf of statesa coordinate 1-cocycle affects a local abelian complex phasetransformation which reflects the local gauge freedom in theconsideration of the phase of a quantum system

From general Cech theory it is well known that every 1-cocycle can be conjugated with a 0-cochain (119905

120572) in the set

1198620(U A) to obtain another equivalent 1-cocycle

1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (10)

If we consider the coboundary operator

Δ0

1198620

(U A) 997888rarr 1198621

(U A) (11)


then the image of Δ0 gives the set of 1-coboundaries of theform Δ0(119905minus1

120572) in 1198611(U A)

Δ0

(119905minus1

120572) = 119905120572sdot 119905minus1

120573 (12)

Thus we obtain that the 1-cocycle 1198921015840120572120573

is equivalent to the 1-cocycle 119892

120572120573in1198851(U A) if and only if there exists a 0-cochain

(119905120572) in the set 1198620(U A) such that

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572) (13)

where the multiplication above is meaningful in the abeliangroup of 1-cocycles 1198851(U A)

Due to the bijective correspondence of line sheaves withcoordinate 1-cocycles with respect to an open coveringU wededuce immediately the following proposition

Proposition 5 The set of isomorphism classes of line sheavesof states over the same topological space of control variables119883denoted by 119868119904119900(L)(119883) is in bijective correspondence with theset of cohomology classes1198671(119883 A)

119868119904119900 (L) (119883) cong 1198671 (119883 A) (14)

The above proposition interpreted in physical termsconstitutes the cohomological formulation of the principle oflocal phase (gauge) invariance in the description of the statespace of a quantum system Furthermore each equivalenceclass [L] equiv L in Iso(L)(119883) has an inverse defined by

Lminus1 = HomA (LA) (15)

where HomA(LA) = Llowast denotes the dual line sheaf of LThis is actually deduced from the fact that we can define thetensor product of two equivalence classes of line sheaves overA so that

LotimesALlowast

cong HomA (L L) equiv EndAL cong A (16)

Hence we obtain the following proposition

Proposition 6 The set of isomorphism classes of line sheavesof states over the same topological space 119883 119868119904119900(L)(119883) hasan abelian group structure with respect to the tensor productover the observable algebra sheaf A and analogously the set ofcohomology classes 1198671(119883 A) is also an abelian group wherethe tensor product of two line sheaves of states corresponds tothe product of their respective coordinate 1-cocycles

23TheHermiticity Condition For reasons implicated by theprobabilistic interpretation of states in quantum mechanicsvia theBorn rule which utilizes the inner product structure ofthe state space we need to focus on the case of isomorphismclasses of Hermitian line sheaves [8 14 24] defined as follows

Definition 7 (i) Given a line sheaf L on 119883 an A-valuedHermitian inner product on L is a skew-A-bilinear sheafmorphism

984858 L oplus L 997888rarr A

984858 (120572119904 120573119905) = 120572 sdot 120573 sdot 984858 (119904 119905)(17)

for any 119904 119905 isin L(119880) 120572 120573 isin A(119880) 119880 open in 119883 Moreover984858(119904 119905) is skew-symmetric namely 984858(119904 119905) = 984858(119904 119905)

(ii) A line sheaf L on 119883 together with an A-valuedHermitian inner product on L constitutes a Hermitian linesheaf

A line sheaf is expressed in local coordinates bijectivelyin terms of a Cech coordinate 1-cocycle 119892

120572120573in 1198851(U A)

associated with the open covering U A Cech coordinate 1-cocycle 119892

120572120573corresponding to a Hermitian line sheaf consists

of local sections of S119880(1A) namely the special unitarygroup sheaf of A of order 1 This is simply a coordinate 1-cocycle 119892

120572120573in 1198851(U A) such that the unitarity condition

|119892120572120573| = 1 is satisfied Clearly in the case that a coordinate

1-cocycle 119892120572120573

is constant we have 119892120572120573

in 1198851(U 119880(1)) orequivalently 119892

120572120573in 1198851(US1)

24 The Exponential Sheaf Sequence and the Role of theIntegers In all cases of physical interest the topological space119883 of control variables is considered to be paracompact whilethe observable algebra sheaf A is a soft sheaf meaning thatevery section over some closed subset in 119883 can be extendedto a section over 119883 More importantly the following shortsequence of abelian group sheaves is exact which modelssheaf-theoretically the process of exponentiation [9]

Proposition 8 The exponential short sequence of abeliangroup sheaves is exact

0 997888rarr Z120580

997888997888rarr Aexp997888997888rarr A 997888rarr 1 (18)

whereZ is the constant abelian group sheaf of integers (sheaf oflocally constant sections valued in the group of integers) suchthat

Ker (exp) = Im (120580) cong Z (19)

Clearly due to the canonical imbedding of the constantabelian group sheaf C into A as well as of C into A we havethe validity of the short exact sequence of constant abeliangroup sheaves


0 997888rarr Z120580

997888997888rarr Cexp997888997888rarr C 997888rarr 1

Ker (exp) = Im (120580) cong Z

(20)

The above exponential short exact sequence can bespecialized further to the following short exact sequence ofconstant abelian group sheaves

0 997888rarr Z120580

997888997888rarr Rexp(2120587119894)997888997888997888997888997888997888rarr U (1) 997888rarr 1 (21)

where R is the constant abelian group sheaf of reals andU(1) is the abelian group sheaf of unit modulus complexes(phases)


The fundamental significance of these three exponentialshort exact sequences of abelian group sheaves cannot beoverestimated In particular we obtain a further refinementof the cohomological formulation of local phase invariancepertaining to the quantum state space according to thefollowing proposition

Theorem 10 Each equivalence class of line sheaves of statesin 119868119904119900(L)(119883) is in bijective correspondence with a cohomologyclass in the integral 2-dimensional cohomology group of 119883

Proof If we consider the exponential short exact sequence ofabelian group sheaves we immediately obtain a long exactsequence in sheaf cohomology which due to paracompact-ness of119883 is reduced to Cech cohomology Thus we have

sdot sdot sdot 997888rarr 1198671

(119883Z) 997888rarr 1198671

(119883A) 997888rarr 1198671

(119883 A)

997888rarr 1198672

(119883Z) 997888rarr 1198672

(119883A) 997888rarr 1198672

(119883 A)

997888rarr sdot sdot sdot

(22)

Because of the fact that A is a soft sheaf we have

1198671

(119883A) = 1198672 (119883A) = 0 (23)

and consequently we obtain

0 997888rarr 1198671

(119883 A)120575119888

997888997888997888rarr 1198672

(119883Z) 997888rarr 0 (24)

Thus equivalently we obtain the following isomorphism ofabelian groups (Chern isomorphism)

120575119888 1198671

(119883 A) cong997888997888997888rarr 1198672

(119883Z) (25)

Since we have shown previously that the set of isomorphismclasses of line sheaves of states over 119883 namely Iso(L)(119883)is in bijective correspondence with the abelian group ofcohomology classes1198671(119883 A) we deduce that

Iso (L) (119883) cong 1198672 (119883Z) (26)

Thus each equivalence class of line sheaves of states is inbijective correspondence with a cohomology class in theintegral 2-dimensional cohomology group of 119883

3 Curvature Differential Invariant andSpectral Beams

31 Sheaves with Potentials We have concluded that a linesheaf L of states on 119883 is classified up to isomorphism bya cohomology class 120575

119888(L) in the abelian group 1198672(119883Z)

A natural question arising in this setting is if it is possibleto express the global invariant information provided by theequivalent line sheavesrsquo classifying integer cohomology classby means of a differential cohomological invariant Thisdifferential invariant should be associated with a paralleltransport rule imposed by a physically induced connectivitystructure on a Hermitian line sheaf of states in analogy

to the original Berry-Simon formulation It is important toemphasize that a particular connection on a line sheaf ofstates provides the means to express this global differentialinvariant locally whereas the latter being a global invariantis independent of the particular means used to represent itlocally Notwithstanding this fact a particular connectionis directed by precise physical requirements associated withthe dynamical evolution of states in relation to a space ofcontrol variables and thus the attainment of this globalinvariant by using local differential means is of physicalsignificance We note that conceptually speaking the notionof a connection on a vector sheaf [9] is rooted on thesheaf-theoretic localization of the homological Kahler-deRham differential mechanism of commutative algebras ofobservables For an elaboration of this viewpoint in relationto physical applications the interested reader may find theliterature instructive [25 26]

Definition 11 A connection nablaE on a vector sheaf of states E isthe following C-linear sheaf morphism

nablaE E 997888rarr Ω1

(A) otimesAE (27)

referring toC-vector space sheaves such that the correspond-ing Leibniz condition is satisfied as follows

nablaE (119886 sdot 119904) = 119886 sdot nablaE (119904) + 119904 otimes 1198890

(119886) (28)

Proposition 12 (i) Every connection nablaE where E is a finiterank-119899 vector sheaf of states on 119883 can be decomposed locallyas follows

nablaE = 1198890

+ 120596 (29)

where 120596 = 120596120572120573

denotes an 119899 times 119899 matrix of sections of local1-forms called the matrix potential of nablaE

(ii) Under a change of local frame matrix 119892 = 119892120572120573

thematrix potentials transform as follows

1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (30)

Proof If we consider a coordinatizing basis of sectionsdefined over an open cover 119880 of119883 denoted by

119890119880

equiv 119880 (119890120572)1le120572le119899

(31)

of the vector sheaf E of rank-119899 called a local frame or a localgauge of E then every continuous local section 119904 isin E(119880)where 119880 isin U can be expressed uniquely with respect to thislocal frame as a superposition

119904 =

119899

sum120572=1

119904120572119890120572

(32)

with coefficients 119904120572inA(119880)The action ofnablaE on these sections

of E is expressed as follows

nablaE (119904) =119899

sum120572=1

(119904120572nablaE (119890120572) + 119890120572 otimes 119889

0

(119904120572))

nablaE (119890120572) =119899

sum120572=1

119890120572otimes 120596120572120573 1 le 120572 120573 le 119899

(33)


where 120596 = 120596120572120573

denotes an 119899 times 119899 matrix of sections of local1-forms Consequently we have

nablaE (119904) =119899

sum120572=1

119890120572otimes (1198890

(119904120572) +

119899

sum120573=1

119904120573120596120572120573)

equiv (1198890

+ 120596) (119904)

(34)

Thus every connection nablaE where E is a finite rank-119899 vectorsheaf on119883 can be decomposed locally as follows

nablaE = 1198890

+ 120596 (35)

In this context nablaE is identified as a covariant derivativeoperator acting on sections of the vector sheaf of statesE and being decomposed locally as a sum consisting ofa flat or integrable part identical with 1198890 and a generallynonintegrable part 120596 called the local frame (gauge) matrixpotential of the connection

The behavior of the local potential 120596 of nablaE under localframe transformations constitutes the ldquotransformation lawof local potentialsrdquo and is obtained as follows Let 119890119880 equiv

119880 119890120572=1119899

and ℎ119881 equiv 119881 ℎ120573=1119899

be two local frames of Eover the open sets 119880 and 119881 of119883 such that 119880 cap 119881 = 0 Let usdenote by 119892 = 119892

120572120573the following change of local framematrix

ℎ120573=

119899

sum120572=1

119892120572120573119890120572 (36)

Under such a local frame transformation 119892120572120573 it is straightfor-

ward to obtain that the local potential 120596 of nablaE transforms asfollows in matrix form

1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (37)

We note that the above holds for any complex vector sheafE We may now specialize in the case of a line sheaf of statesL equipped with a connection denoted by the pair (L nabla) Inthis case of interest due to the isomorphism

LotimesALlowast


we obtain that the local form of a connection over an open setis just a local 1-form or local potential (ie a local continuoussection ofΩ1)

Definition 13 A line sheaf of states L equippedwith a connec-tion nabla is called a line sheaf with potentials or equivalently adifferential line sheaf and denoted by the pair (L nabla)

32 Unitary Rays and Their Cohomological CharacterizationWe recall that a line sheaf of states L on119883 together with anA-valuedHermitian inner product onL constitutes aHermitianline sheaf

Definition 14 A connection nabla on L is called Hermitian if it iscompatible with 984858

1198890

984858 (119904 119905) = 984858 (nabla119904 119905) + 984858 (119904 nabla119905) (39)

for any 119904 119905 isin L(119880) 119880 open in119883

Definition 15 A Hermitian differential line sheaf or equiva-lently a unitary ray denoted by (L nabla 984858) is a Hermitian linesheaf equipped with a Hermitian connection

First we note that although we use the same symbol 984858 inthe second part of the above it refers to an extension of theA-valued Hermitian inner product on L defined as follows

984858 Ω (L) oplus L 997888rarr Ω(L)

984858 (119904 otimes 119905 1199041015840

) = 984858 (119904 1199041015840

) sdot 119905(40)

for any 119904 1199041015840 isin L(119880) 119905 isin Ω(119880) and119880 open in119883 FurthermoreΩ is considered to be a vector sheaf on119883

Second if we have a Hermitian line sheaf (L 984858) whichis also equipped with a connection nabla described in terms oflocal potentials 120596 = (120596

120572) it is straightforward to see that the

compatibility condition of the connectionwith theHermitianinner product 984858 is satisfied such that (L nabla 984858) is a unitary rayif and only if

120596 + 120596 = 1198890

(984858) (41)

where 1198890 denotes the logarithmic universal derivation of theobservable abelian group sheaf A to the abelian group sheaf of1-formsΩ1 that is the universal morphism of abelian groupsheaves

1198890

A 997888rarr Ω1 (42)

defined by the relation

1198890

(119906) = 119906minus1

sdot 1198890

(119906) (43)

for any continuous invertible local section 119906 isin A(119880) with 119880open set in 119883 Given the validity of the Poincare Lemma wealso have

Ker (1198890) = C (44)

Third by considering the Hermitian line sheaf (L 984858) anda local frame of L we may apply locally the Gram-Schmidtorthonormalization procedure in our context such that Lhas locally an orthonormal frame Thus with respect to thisorthonormal frame of L we obtain

1198890

(984858) = 0 (45)

and finally we deduce that 120596 = minus120596Consequently given that everyHermitian line sheaf (L 984858)

admits a Hermitian connection we have specified completelythe notion of a unitary ray denoted by (L nabla 984858) according tothe above In order to simplify further the notation we denotea unitary ray namely a Hermitian differential line sheaf by(L nabla984858) The next important task is to establish the local form

of a unitary ray with respect to an open covering of 119883 andinversely determine the conditions that local componentshave to satisfy so that they constitute a unitary ray


Proposition 16 (i)The local form of a differential line sheaf isgiven by

(L nabla) larrrarr (119892120572120573 120596120572) isin 1198851

(U A) times 1198620 (U Ω1) (46)

(ii) An arbitrary pair (119892120572120573 120596120572) isin 1198851(U A) times 1198620(U Ω1)

determines a differential line sheaf if

1205750

(120596120572) = 1198890

(119892120572120573) (47)

Proof A line sheaf is expressed in local coordinates bijectivelyin terms of a Cech coordinate 1-cocycle 119892

120572120573in 1198851(U A)

associated with the coveringU A connection nabla is expressedbijectively in terms of a 0-cochain of 1-forms called the localpotentials of the connection and denoted by 120596

120572with respect

to the covering U of 119883 that is 120596120572isin 1198620(U Ω1) We deduce

that the local form of a differential line sheaf is as follows


(U A) times 1198620 (U Ω1) (48)

Conversely an arbitrary pair (119892120572120573 120596120572) isin 1198851(U A) times

1198620

(U Ω1) determines a differential line sheaf if the ldquotrans-formation law of local potentialsrdquo is satisfied by this pair thatis

120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)

Thus given an open covering U = 119880120572 a 0-cochain

(120596120572) valued in the sheaf Ω1 determines the local form of a

connectionnabla on the line sheaf L where the latter is expressedin local coordinates bijectively in terms of a Cech coordinate1-cocycle (119892

120572120573) valued in A with respect to U if and only

if the corresponding local 1-forms 120596120572of the 0-cochain with

respect to U are pairwise intertransformable (locally gauge-equivalent) on overlaps 119880

120572120573via the local frame transition

functions (isomorphisms) 119892120572120573

isin A(119880120572120573) according to the

ldquotransformation law of local potentialsrdquo

1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)

and 1205750 denotes the 0th coboundary operator 1205750

1198620(U Ω1) rarr 1198621(U Ω1) such that1205750

(120596120572) = 120596120573minus 120596120572 (52)

As a straightforward application of the above propositionwe obtain the following corollary

Proposition 17 (i) The local form of a unitary ray is given by


(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1


determines a unitary ray if

1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)

Proof A Cech coordinate 1-cocycle 119892120572120573

of a unitary rayconsists of local sections of S119880(1A) namely the specialunitary group sheaf ofA of order 1This is simply a coordinate1-cocycle 119892



1-cocycle 119892120572120573



120572120573in 1198851(US1) Moreover given a Hermitian

line sheaf (L 984858) which is also equipped with a connectionnabla described in terms of local potentials 120596 = (120596

120572) isin

1198620(U Ω1) the compatibility condition of the connectionwith the Hermitian inner product 984858 is satisfied if and only if120596 + 120596 = 1198890(984858)

33 Gauge Equivalence of Unitary Rays We consider two linesheaves which are equivalent via an isomorphism ℎ L cong997888rarr L1015840Wewould like to extend the notion of equivalence for two linesheaves equipped with a connective structure namely twodifferential line sheaves (L nabla) and (L1015840 nabla1015840) and in particulartwo unitary rays

Definition 18 Given an isomorphism ℎ L cong997888rarr L1015840 of linesheaves of states we say that nabla is frame or gauge equivalentto nabla1015840 if they are conjugate connections under the action of ℎ

nabla1015840

= ℎ sdot nabla sdot ℎminus1

(55)

Thus we may consider the set of equivalence classes ondifferential line sheaves (L nabla) under an isomorphism ℎ aspreviously denoted by Iso(L nabla) It is now important to findthe relation between Iso(L nabla) and the abelian group Iso(L)For this purpose we need to utilize the local form of adifferential line sheaf (L nabla) and concomitantly the local formof a unitary ray (L nabla

984858)

Next we consider two line sheaves which are equivalentvia an isomorphism ℎ L cong

997888rarr L1015840 such that theircorresponding connections are conjugate under the action ofℎ

nabla1015840

= ℎnablaℎminus1

(56)

Under these conditions the differential line sheaves (L nabla)and (L1015840 nabla1015840) are called gauge or frame equivalent Thus wemay consider the set of gauge equivalence classes [(L nabla)] ofdifferential line sheaves as above denoted by Iso(L nabla)

Proposition 19 The set of gauge equivalence classes of unitaryrays 119868119904119900(L nabla

984858) is an abelian subgroup of the set of gauge

equivalence classes of differential line sheaves 119868119904119900(L nabla) whichin turn is an abelian subgroup of the abelian group 119868119904119900(L)


Proof If we consider the local form of the tensor product oftwo gauge equivalent differential line sheaves we have

(L nabla) otimesA (L1015840

nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)

which satisfies the ldquotransformation law of local potentialsrdquo

1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)

Moreover the inverse of a pair (119892120572120573 120596120572) is given by

(119892minus1120572120573 minus120596120572) whereas the neutral element in the group Iso(L nabla)

is given by (119894119889120572120573 0) which corresponds to the trivial dif-

ferential line sheaf (A 1198890) Clearly since a unitary ray is adifferential line sheaf such that |119892

120572120573| = 1 the set of gauge

equivalence classes of unitary rays Iso(L nabla984858) is an abelian

subgroup of the set of gauge equivalence classes of differentialline sheaves Iso(L nabla)

Next we need to specify the local conditions under whichpairs of the form (119892

120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge

equivalent differential line sheaves and concomitantly gaugeequivalent unitary rays

Proposition 20 For any two gauge equivalent differential linesheaves (L nabla) and (L1015840 nabla1015840) in 119868119904119900(L nabla) the following holds(L nabla) is equivalent to (L1015840 nabla1015840) meaning that ℎ L cong997888rarr L1015840and nabla1015840 = ℎnablaℎminus1 if and only if there exists a 0-cochain (119905

119886) isin

1198620(U A) = prod120572A(119880120572) such that

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)

where by definition the coordinate 1-cocycles of the line sheavesL and L1015840 associated with the common open covering U aregiven by 119892

120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin

1198851(U A) according to the following diagram

119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)

Proof It is clear that an 1-cocycle 1198921015840120572120573

is equivalent to another1-cocycle 119892

120572120573in 1198851(U A) if and only if there exists a 0-

cochain (119905120572) in the set 1198620(U A) such that

1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573

under conjugation with (119905120572)

This is equivalently written as an action of the image of

the coboundary operator that is of the abelian group of 1-coboundaries in 1198611(U A) on the abelian group of 1-cocyclesin 1198851(U A)

Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)

Thus the hypothesis of an isomorphism ℎ L cong997888rarr L1015840 isequivalent to the existence of a 0-cochain (119905

119886) isin 1198620(U A) =

prod120572A(119880120572) such that the above condition is satisfied At a

further stage the gauge equivalence of the differential linesheaves (L nabla) and (L1015840 nabla1015840) induced by ℎ harr (119905

120572) implies that

nabla1015840

= ℎnablaℎminus1 Hence by using the local expressions of the

connections nabla1015840 and nabla by means of the local 1-forms 1205961015840120572and

120596120572correspondingly we immediately obtain

1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)

Thus we have shown that

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)

34 De Rham Cohomology Class of the Curvature The possi-bility of expressing the classifying integer cohomology classof isomorphic line sheaves of states by means of a classifyingdifferential invariant of isomorphic differential line sheavesor unitary rays would provide a cohomological refinement ofthe physical gauge principle according to which local gaugeinvariance implies the existence of local gauge potentials andthus of a connection on a line sheaf of states Given thatlocal gauge invariance is depicted by means of coordinate 1-cocycles with respect to a covering U of 119883 and the fact thatthe localization of a connection is expressed in terms of localgauge potentials with respect to U it is natural to seek for arelation between the invariant of the former that is an integercohomology class and the differential invariant of the latterthat is the class of the curvature of the connection After theseintroductory remarks we briefly recall the following well-known facts

The universal C-derivation of the observable algebrasheaf A to the universal A-module sheaf Ω1(A) = Ω1 is theuniversal C-linear sheaf morphism 1198890 A rarr Ω1 such thatthe Leibniz condition 1198890(119904 sdot 119905) = 119904 sdot 1198890(119905) + 119905 sdot 1198890(119904) is satisfiedfor any continuous local sections 119904 119905 belonging toA(119880) with119880 open set in119883 Moreover given the validity of the PoincareLemma Ker(1198890) = C and the fact that A is a soft observablealgebra sheaf the sequence ofC-vector space sheaves is exact

0 997888rarr C 997888rarr A 997888rarr Ω1

(A) 997888rarr sdot sdot sdot 997888rarr Ω119899

(A)

997888rarr sdot sdot sdot (65)


Therefore the sequence of C-linear sheaf morphisms

A 997888rarr Ω1


(A) 997888rarr sdot sdot sdot (66)

is a complex of C-vector space sheaves called the sheaf-theoretic de Rham complex of A Thus the sheaf-theoretic deRham complex of the observable algebra sheaf A constitutesa resolution of the constant sheaf C

If we assume that the pair (E nablaE) denotes a complexvector sheaf of states equipped with a connective structuredefined by a connection nablaE on E then nablaE induces a sequenceof C-linear morphisms

E 997888rarr Ω1

(A) otimesAE 997888rarr sdot sdot sdot 997888rarr Ω119899

(A) otimesAE 997888rarr sdot sdot sdot (67)

or equivalently

E 997888rarr Ω1

(E) 997888rarr sdot sdot sdot 997888rarr Ω119899

(E) 997888rarr sdot sdot sdot (68)

where the morphism

nabla119899

Ω119899

(A) otimesAE 997888rarr Ω119899+1

(A) otimesAE (69)

is given by the formula

nabla119899

(120596 otimes V) = 119889119899 (120596) otimes V + (minus1)119899 120596 and nabla (V) (70)

for all 120596 isin Ω119899(A) V isin E It is immediate to see that nabla0 = nablaE

Definition 21 Thecomposition ofC-linearmorphismsnabla1∘nabla0is called the curvature of the connection nablaE

nabla1

∘ nabla0

= Rnabla E 997888rarr Ω

2

(A) otimesAE = Ω2

(E) (71)

As a straightforward consequence of the above definitionwe obtain the following

Proposition 22 The curvature Rnablaof a connection nablaE on the

vector sheaf of states E is an A-linear sheaf morphism that isa A-covariant or equivalently an A-tensor

The A-covariant nature of the curvature Rnablais to be

contrasted with the connectionnablaE which is onlyC-covariantFrom the above we directly derive the following corollary

Proposition 23 The sequence of C-linear sheaf morphisms

E 997888rarr Ω1



is a complex of C-vector space sheaves if and only if

Rnabla= 0 (73)

Thus the curvature A-linear sheaf morphism Rnabla

expresses the obstruction for the above sequence to qualifyas a complex We say that the connection nablaE is integrable ifRnabla= 0 and we refer to the obtained complex as the sheaf-

theoretic de Rham complex of the integrable connectionnablaE on the vector sheaf E in that case It is also usual to calla connection nablaE flat if R

nabla= 0 We note that the universal

C-derivation 1198890 onA defines an integrable or flat connection

Furthermore we need to derive the local form of thecurvature R

nablaof a connection nablaE where E is a vector sheaf

E on 119883 Due to its property of A-covariance a nonvanishingcurvature represents in this context the A-covariant andthus geometrically observable deviation from the unob-structed or monodromic form of variation correspondingto an integrable connection Moreover since the curvatureRnablais an A-linear morphism of sheaves of A-modules that

is an A-tensor Rnablamay be thought of as an element of

End(E)otimesAΩ2(A) = Ω2(End(E)) or equivalently Rnabla

isin

Ω2(End(E)) Hence the local form of the curvature Rnablaof

a connection nablaE consists of local 119899 times 119899 matrices having forentries local 2-forms In particular the local form of thecurvature R

nabla|119880 where 119880 is open in 119883 in terms of the local

potentials 120596 is expressed by

Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)

as it can be easily shown by substitution of the local potentialsin the compositionnabla1 ∘nabla0 Furthermore by application of thedifferential operator 1198892 on the above we obtain

1198892 Rnabla

1003816100381610038161003816119880 = Rnabla

1003816100381610038161003816119880 and 120596 minus 120596 and Rnabla

1003816100381610038161003816119880 (75)

The behavior of the curvature Rnablaof a connection nablaE

under local frame transformations constitutes the ldquotransfor-mation law of potentialsrsquo strengthrdquo If we agree that 119892 = 119892

120572120573

denotes the change of local frame matrix we have previouslyconsidered in the discussion of the transformation law oflocal connection potentials we deduce the following localtransformation law

Rnabla

119892

997891997888rarr R1015840nabla= 119892minus1

(Rnabla) 119892 (76)

that is they transform covariantly by conjugation withrespect to a local frame transformation It is useful to collectthe above findings in the form of the following proposition

Proposition 24 (i) The local form of the curvature Rnabla|119880

where 119880 is open in 119883 in terms of the local matrix potentials120596 is given by

Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)


the localform of the curvature transforms by conjugation

Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast


we obtain the following simplifications the local form of aconnection over an open set is just a local 1-form or localpotential (ie a local continuous section of Ω1) whence thelocal form of the curvature of the connection over an open set


is a local 2-form The significant result obtained by the localtransformation law in this case is that the curvature is localframe invariant that is it does not change under any localframe (gauge) transformation

Rnabla

119892

997891997888rarr R1015840nabla= Rnabla= 1198891

120596 (80)

Thus we obtain a global 2-form Rnabladefined over 119883 which is

also a closed 2-form because of the fact that

1198892Rnabla= 0 (81)

Hence we obtain the following

Proposition 25 For a line sheaf of states L equipped with aconnection nabla or equivalently for a differential line sheaf (L nabla)the curvature R

nablaof the connection is a global closed 2-form

Clearly the same holds true if we restrict a differential linesheaf to a unitary ray

Proposition 26 The curvature R determines a global differ-ential invariant of gauge equivalent differential line sheavesor gauge equivalent unitary rays in terms of its de Rhamcohomology class [R]

Proof An immediate important consequence of the charac-terization of gauge equivalent differential line sheaves in localterms with respect to an open covering U is that all of themhave the same curvature Indeed by applying the differentialoperator 1198891 on the above relation and using the fact that

1198891

∘ 1198890

= 0 (82)

from which we have that

1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)

Hence any two gauge equivalent differential line sheaves (andthus unitary rays) always have the same curvature denoted byR We remind that R is a global 2-form on 119883 since it is localframe-change invariant which is also closed because of thefact that

1198892

∘ 1198891

120596120572= 1198892R = 0 (85)

Thus the global 2-form R which belongs to Ker(1198892)

Ω2 rarr Ω3 called Ω2119888 being a C-vector sheaf subspace

of Ω2 determines a global differential invariant of gaugeequivalent differential line sheaves This is the case becausethe global 2-form R determines a 2-dimensional de Rhamcohomology class [R] identified as a 2-dimensional complexCech cohomology class in 119867

2(119883C) In particular if weconsider a differential line sheaf the differential invariant deRhamcohomology class [R] is independent of the connectionused to represent R locally In other words a particularconnection of a differential line sheaf (or a unitary ray)provides themeans to express this global differential invariantlocally whereas the latter is independent of the particularmeans used to represent it locally

35TheCurvature Recognition IntegralityTheorem Since anytwo gauge equivalent differential line sheaves of states (orunitary rays) have the same curvature we conclude that theyare physically indistinguishable This means that the abeliangroup Iso(L nabla) is partitioned into orbits over the image ofIso(L nabla) into Ω2

119888 where each orbit (fiber) is labelled by a

closed 2-form R ofΩ2119888

Iso (L nabla) = sumRIso (L nabla)R (86)

Thus the abelian group of equivalence classes of differentialline sheaves fibers over those elements of Ω2

119888 namely over

those closed global 2-forms in Ω2119888which can be identified as

curvatures In this way a pertinent problem is the concreteidentification of the image of Iso(L nabla) intoΩ2

119888 Put differently

we are looking for an intrinsic characterization of those globalclosed 2-forms in Ω2

119888 which are instantiated as curvatures

of gauge equivalence classes of differential line sheaves (orunitary rays) The resolution of this problem is provided bythe sheaf-theoretic formulation of the Chern-Weil integralitytheorem [9] According to this the de Rham cohomologyclass [R] of any differential line sheaf in 1198672(119883C) is in theimage of a cohomology class in the integral 2-dimensionalcohomology group1198672(119883Z) into1198672(119883C)

Theorem 27 A global closed 2-form is the curvature R of adifferential line sheaf if and only if its 2-dimensional de Rhamcohomology class is integral namely [R] isin Im(1198672(119883Z) rarr1198672(119883C))

Proof We know that each equivalence class of line sheavesis in bijective correspondence with a cohomology class inthe integral 2-dimensional cohomology group of119883 Thus wehave to show that a 2-dimensional deRhamcohomology classis a curvature differential invariant class of gauge equivalentdifferential line sheaves if and only if it is an integral 2-dimensional cohomology class First by the natural injectionZ 997893rarr C we obtain

1198672

(119883Z) 997888rarr 1198672

(119883C) (87)

where any cohomology class belonging to the image of theabove map is called an integral cohomology class Next weconsider the following exact sequences

0 997888rarr Z120580


which is the exponential short exact sequence of abeliangroup sheaves such that

Ker (exp) = Im (120580) cong Z (89)

and the short exact sequence of C-vector sheaves

0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)

which is a fragment of the de Rham resolution of the constantsheaf C such that

Ker (1198890) = Im (120576) cong C (91)


Furthermore we consider the following commutative dia-gram

119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)

Thus we obtain the relation

2120587119894 sdot 1198890

= 1198890

∘ exp (93)

If we take the relevant fragments of the corresponding longexact sequences in cohomology we obtain the followingcommutative diagram

H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)

Thus the image of a cohomology class of 1198672(119883Z) into anintegral cohomology class of 1198672(119883C) corresponds to thecohomology class specified by the image of the 1-cocycle 119892

120572120573

into1198671(119883 1198890A) namely by the 1-cocycle (12120587119894)1198890(119892120572120573)

In more detail we first have that due to the exactness ofthe exponential sheaf sequence of abelian group sheaves

0 997888rarr Z120580

997888997888rarr Aexp997888997888rarr A 997888rarr 1

119892120572120573= exp (119908

120572120573)

(95)

where (119908120572120573) isin 1198621(UA) as well as

120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)

Explicitly we may consider 119908120572120573= ln(119892

120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)

Now we need to locate the curvature 2-dimensional complexde Rham cohomology class differential invariant namely thecomplex 2-dimensional cohomology class [R] in 1198672(119883C)For this purpose regarding the curvature we have that

R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)

By the transformation law of local potentials it holds

1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)

and 119892120572120573isin 1198851(U A) whence 1205750 denotes the 0th coboundary

operator 1205750 1198620(U Ω1) rarr 1198621(U Ω1) such that

1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)

where 1198890(119908120572120573) isin 1198851(U 1198890A) Therefore we deduce that

1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)

If we inspect the short exact sequence of C-vector sheaves

0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)

we immediately deduce that

120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)

Thus finally we obtain

[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)

Thus we have obtained an intrinsic characterization of thesubset of those global closed 2-forms in Ω2

119888 which are

instantiated as curvatures of gauge equivalence classes ofdifferential line sheaves denoted by Ω2

119888Z Therefore a globalclosed 2-form is the curvature R of a differential line sheaf ifand only if its 2-dimensional de Rham cohomology class isintegral namely [R] isin Im(1198672(119883Z) rarr 1198672(119883C))

4 Spectral [R]-Beams and the Role of theFundamental Group

41 Free Action of 1198671(119883S1) on Spectral [R]-Beams Fromthe curvature recognition integrality theorem we have con-cluded that the abelian group Iso(L nabla) is partitioned intoorbits overΩ2

119888Z where each orbit (fiber) is labelled by an inte-gral global closed 2-form R ofΩ2

119888Z providing the differentialinvariant [R] of this orbit in de Rham cohomology

Iso (L nabla) = sum

RisinΩ2119888Z

Iso (L nabla)R (107)


If we restrict the abelian group Iso(L nabla) to gauge equivalentHermitian differential line sheaves (unitary rays) we obtainan abelian subgroup of the former denoted by Iso(L nabla

984858)

Clearly the latter abelian group is also partitioned into orbitsover Ω2

119888Z where each orbit is labelled by an integral globalclosed 2-form R of Ω2

119888Z where R is the curvature of thecorresponding gauge equivalence class of unitary rays

Definition 28 We call each Hermitian differential line sheaf(L nabla984858) belonging to an orbit Iso(L nabla

984858)R a unitary R-ray

whence the orbit itself is called a spectral [R]-beam and ischaracterized by the integral differential invariant [R]

Each spectral [R]-beam consists of gauge equivalent uni-taryR-rays which are indistinguishable from the perspectiveof their common curvature integral differential invariant[R] A natural question arising in the context of gaugeequivalent unitary R-rays is how they are related to eachother In other words although all gauge equivalent unitaryR-rays cannot be distinguished from the perspective of theircurvature differential invariant is there any other intrinsicway that we can distinguish among them From a quantumphysical perspective if such an intrinsic and invariant way ofdistinguishing among gauge equivalent unitaryR-rays existsit means that there exists a global kinematical symmetry ofspectral [R]-beams

Proposition 29 There exists a free group action of theabelian group 1198671(119883S1) on the abelian group of unitary rays119868119904119900(L nabla

984858) where S1 denotes the abelian group sheaf S1 equiv

U(1) equiv S119880(1C) which is restricted to a free group action oneach spectral [R]-beam

Proof There exists a free group action of S1 997893rarr C 997893rarr A onthe group sheaf A of invertible elements of A

S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)

with 120585 isin S1 and 119891 isin A(119880) for any open 119880 in 119883 This actionis transferred naturally as a free action to the correspondinggroups of coordinate 1-cocycles of the respective abeliangroup sheaves

1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free

action can be also extended to the corresponding cohomol-ogy groups being still a free action

1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)

Now we can finally define a group action of1198671(119883S1) on the

abelian group Iso(L nabla) as follows We consider 120585 equiv [(120585120572120573)] isin

1198671(119883S1) [(L nabla)] isin Iso(L nabla) and we define the soughtgroup action as follows

120585 sdot [(L nabla)] = [(120585 sdot L nabla)] equiv [(L1015840 nabla)]

L1015840 = 120585 sdot Llarrrarr (120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(111)

It is immediate to show that the pair (120585120572120573sdot 119892120572120573 120596120572) actually

satisfies the ldquotransformation law of local potentialsrdquo that is

120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)

as a consequence of the Poincare Lemma we can easily showthat the above defined group action of 1198671(119883S1) on theabelian group Iso(L nabla) is actually free

For this purpose we consider the equivalent differentialline sheaves (L nabla) and (L1015840 nabla) = 120585 sdot [(L nabla)] = [(120585 sdotL nabla)] in theabelian group Iso(L nabla) where 120585 = [(120585

120572120573)] isin 1198671(119883S1) and

thus (120585120572120573) isin 1198851(US1) according to the above Due to the

above equivalence we conclude that there exists a 0-cochain(119905119886) isin 1198620(U A) = prod

120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)

and hence equivalently

[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)

where (120585120572120573) isin 1198851(US1) 997893rarr (120585

120572120573) isin 1198851(U A) Furthermore

by hypothesis we have

1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)

such that 119905119886isin Ker(1198890) = C Thus we obtain that an

automorphism ℎ of a line sheaf L is also an automorphism ofthe differential line sheaf (L nabla) if and only if 119905

119886isin Ker(1198890) =

C Hence in the unitary case considered we deduce that the0-cochain (119905

119886) isin 119862

0

(U A) actually belongs to 1198620(US1)Therefore we conclude that

[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)

which proves the freeness of the above actionWe note that if we do not assume the unitarity condition

the same argument shows that the group action of1198671(119883 C)induced by the natural injection C 997893rarr A on the abelian groupIso(L nabla) is actually free where in this more general case wehave that [(120585

120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)

Consequently the free group action of 1198671(119883S1) onIso(L nabla) is restricted to a free group action on its abelian


subgroup of unitary rays Iso(L nabla984858) Since the latter abelian

group is partitioned into spectral [R]-beams over Ω2119888Z we

conclude that the above free group action is finally transferredas a free group action of 1198671(119883S1) on each spectral [R]-beam

Definition 30 A cohomology class in the abelian group1198671(119883S1) is called a polarization phase germ of a spectral[R]-beam

The intuition behind the above definition is that a coho-mology class in the abelian group 1198671(119883S1) cong 1198671(119883119880(1))can be evaluated at a homology cycle 120574 isin 119867

1(119883) by means of

the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)

to obtain a global observable gauge-invariant phase factorin 119880(1) Thus gauge equivalent unitary R-rays may beintrinsically distinguished by means of a polarization phasegerm identified as a cohomology class in1198671(119883S1)

If we assume that the underlying space 119883 of controlvariables is a locally path-connected space thenwe obtain thefollowing

Proposition 31 A polarization phase germ of a spectral [R]-beam is realized by a representation of the fundamental groupof the topological space 119883 of control variables to S1

Proof As an immediate consequence of the Hurewicz iso-morphism we obtain

Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)

By restriction to the unitary case we obtain

Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)

Thus a polarization phase germ expressed in terms of acohomology class in1198671(119883S1) is realized by a representationof the fundamental group of the topological space 119883 toS1

42 Affine Space Structure of Spectral [R]-Beams We haveshown that there exists a free group action of 1198671(119883S1) congHom(120587

1(119883)S1) on each spectral [R]-beam A natural prob-

lem in this context is the specification of the appropriateconditions whichwould qualify this free action as a transitiveone as well

For this purpose we consider the following sequence ofabelian group sheaves

1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)

As a consequence of the Poincare Lemma we have that

Ker (1198890) = C (123)

Moreover we have that 1198891 ∘ 1198890 = 0 and thus Im(1198890) subeKer(1198891) Now we consider those closed 1-forms 120579

120572ofΩ1 for

which the following condition is satisfied

Im (1198890

) = Ker (1198891) (124)

Definition 32 The closed 1-forms 120579120572

of Ω1 satisfyingIm(1198890) = Ker(1198891) are called logarithmically exact closed 1-forms

Proposition 33 The following sequence of abelian groupsheaves is an exact sequence if restricted to logarithmicallyexact closed 1-forms

1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)

Theorem 34 A spectral [R]-beam is an affine space withstructure group of the characters of the fundamental groupwithrespect to logarithmically exact closed 1-forms

Proof By the exact sequence of abelian group sheaves ofthe previous proposition we obtain a 0-cochain (120579

120572) of

logarithmically exact closed 1-forms

(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)

Hence for a 0-cochain (120579120572) of logarithmically exact closed 1-

forms 120579120572 there exists a 0-cochain 119905

120572in A such that

120579120572= 1198890

(119905minus1

120572) (127)

This 0-cochain (120579120572) may be considered as representing an

integrable connection nabla on a line sheaf K whose coordinate1-cocycle with respect to an open coveringU is given by

120577120572120573= 119905minus1

120573119905120572 (128)

Clearly in this case the transformation laws of local potentialsare satisfied as follows

1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)

Next we consider a spectral [R]-beam namely an orbitIso(L nabla

984858)R consisting of gauge equivalent unitary R-rays

which are indistinguishable from the perspective of theircommon differential invariant [R] Again we also considerthose closed 1-forms 120579

120572of Ω1 for which the following

condition is satisfied

Im (1198890

) = Ker (1198891) (130)

namely the logarithmically exact closed 1-forms We take apair of equivalent unitary R-rays denoted by (L nabla

984858) (L1015840 nabla1015840

984858)

correspondingly Then we have that

R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)


We conclude that (120596120572minus1205961015840120572) is of the form 120579

120572ofΩ1 namely it

is a logarithmically exact closed 1-form This means that weobtain a 0-cochain (120596

120572minus 1205961015840120572) of logarithmically exact closed

1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)

Hence for a 0-cochain (120596120572minus 1205961015840120572) of logarithmically exact

closed 1-forms there exists a 0-cochain 119905120572in A such that

120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)

Furthermore we may consider another unitary R-ray inthe same orbit denoted by (L10158401015840 nabla10158401015840

984858) characterized locally as

follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)

Thus if we consider any two unitary R-rays (L nabla) (L1015840 nabla1015840984858)

we can always find another unitary R-ray (L10158401015840 nabla10158401015840984858= nabla984858)

equivalent to both of them namely belonging to the sameorbit over their common differential invariant [R] or equiv-alently belonging to the same spectral [R]-beam Thereforefor any two unitary R-rays (L nabla

984858) (L1015840 nabla1015840

984858) we can always

substitute for instance the second one of them (L1015840 nabla1015840984858) with

an equivalent unitary R-ray characterized by the same 0-cochain of potentials like the first one namely by (L10158401015840 nabla

984858)

whose local form is given by the above defined pair (11989210158401015840120572120573 120596120572)

Consequently we obtain

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)

Thus (11989210158401015840120572120573sdot 119892minus1120572120573) isin 1198851(UKer(1198890)) or equivalently by the

Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)

which in the quantum unitary case considered reduces to

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)

The above can be equivalently formulated as follows

11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573

isin 1198851(US1) By considering the correspondingcohomology classes we obtain

[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)

Thus we finally deduce that

[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)

where [(120585)] = [(120585120572120573)] isin 1198671(119883S1) Therefore we finally

arrive at the following conclusionThe free group action of 1198671(119883S1) on a spectral [R]-

beam is also transitive with respect to logarithmically exactclosed 1-forms and therefore a spectral [R]-beam becomesa 1198671(119883S1) cong Hom(120587

1(119883)S1)-affine space or equivalently

an affine space with structure group the characters of thefundamental group

Thus each partition block or fiber Iso(L nabla984858)R labelled by

the curvature differential invariant [R] namely each spectral[R]-beam is an affine spacewith structure group1198671(119883S1) congHom(120587

1(119883)S1) In this manner any two unitary R-rays

differ by an element of 1198671(119883S1) and conversely anytwo unitary rays which differ by an element of 1198671(119883S1)are characterized by the same differential invariant [R] orequivalently areR-rays of the same spectral [R]-beamHencealthough all gauge equivalent unitary R-rays cannot be dis-tinguished from the perspective of their common curvaturedifferential invariant there exists a free and transitive actionof the group 1198671(119883S1) cong Hom(120587

1(119883)S1) characterized

as the global kinematical symmetry group of a [R]-beamwhich completely distinguishes among them by means ofcharacters of the fundamental group of 119883 Inversely fromany one unitary R-ray we can obtain intrinsically its wholeequivalence class by means of the free and transitive actionof the abelian group 1198671(119883S1) on the depicted one Weconclude that whenever two unitary rays are characterizedby the same differential invariant [R] namely they belongto the same equivalence class (orbit) under the action of1198671

(119883S1) on Iso(L nabla984858)R (which is actually the only class

due to transitivity of this action identified as a spectral [R]-beam) and then they differ by a character of the fundamentalgroup of119883

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Thepaper is dedicated to thememory of Professor AnastasiosMallios who has always insisted on the important physical


role of differential vector sheaves in quantum theory andhighlighted the essentially cohomological character of gaugetheory

References

[1] M V Berry ldquoQuantal phase factors accompanying adiabaticchangesrdquo Proceedings of the Royal Society London Series AMathematical Physical and Engineering Sciences vol 392 no1802 pp 45ndash57 1984

[2] B Simon ldquoHolonomy the quantum adiabatic theorem andBerryrsquos phaserdquo Physical Review Letters vol 51 no 24 pp 2167ndash2170 1983

[3] Y Aharonov and D Bohm ldquoSignificance of electromagneticpotentials in the quantum theoryrdquo vol 115 pp 485ndash491 1959

[4] Y Aharonov and D Bohm ldquoFurther considerations on electro-magnetic potentials in the quantum theoryrdquo Physical Reviewvol 123 pp 1511ndash1524 1961

[5] F Wilczek and A Shapere Geometric Phases in Physics WorldScientific 1989

[6] M V Berry ldquoPancharatnam virtuoso of the Poincare sphererdquoCurrent Science vol 67 no 4 pp 220ndash223 1994

[7] A Bohm A Mostafazadeh H Koizumi Q Niu and JZwanzigerTheGeometric Phase in Quantum Systems Texts andMonographs in Physics Springer Berlin Germany 2003

[8] AMallios ldquoGeometry of vector sheaves an axiomatic approachto differential geometryrdquo inVector Sheaves GeneralTheory vol1 Kluwer Academic Publishers Dordrecht The Netherlands1998

[9] A MalliosGeometry of Vector Sheaves An Axiomatic Approachto Differential Geometry Vol II Geometry Examples and Appli-cations vol 439 of Mathematics and Its Applications KluwerAcademic Publishers Dordrecht The Netherlands 1998

[10] G E Bredon Topology and Geometry Springer New York NYUSA 1993

[11] G E Bredon Sheaf Theory vol 170 of Graduate Texts inMathematics Springer Berlin Germany 2nd edition 1997

[12] E Vassiliou Geometry of Principal Sheaves Kluwer AcademicDodrecht The Netherlands 2004

[13] F Hirzebruch Topological Methods in Algebraic GeometryClassics in Mathematics Springer Berlin Germany 1995

[14] S S Chern Complex Manifolds without Potential TheorySpringer New York NY USA 1979

[15] A Mallios Modern Differential Geometry in Gauge TheoriesVol 1 Maxwell Fields Birkhauser Boston Mass USA 2006

[16] A Mallios Modern Differential Geometry in Gauge TheoriesYang-Mills Fields vol 2 Birkhauser Boston Mass USA 2009

[17] S A Selesnick ldquoLine bundles and harmonic analysis oncompact groupsrdquo Mathematische Zeitschrift vol 146 no 1 pp53ndash67 1976

[18] E Zafiris ldquoGeneralized topological covering systems on quan-tum eventsrsquo structuresrdquo Journal of Physics A Mathematical andGeneral vol 39 no 6 pp 1485ndash1505 2006

[19] E Zafiris ldquoSheaf-theoretic representation of quantum measurealgebrasrdquo Journal of Mathematical Physics vol 47 no 9 ArticleID 092103 2006

[20] M Epperson andE ZafirisFoundations of Relational RealismATopological Approach toQuantumMechanics and the Philosophyof Nature Lexington Books Lanham Md USA 2013

[21] A Mallios ldquoOn localizing topological algebrasrdquo ContemporaryMathematics vol 341 p 79 2004

[22] R Resta ldquoManifestations of Berryrsquos phase in molecules andcondensedmatterrdquo Journal of Physics CondensedMatter vol 12no 9 pp R107ndashR143 2000

[23] A Mallios Topological Algebras Selected Topics North-Holland Amsterdam The Netherlands 1986

[24] P A Griffiths ldquoHermitian differential geometry and the theoryof positive and ample holomorphic vector bundlesrdquo Journal ofMathematics and Mechanics vol 14 pp 117ndash140 1965

[25] A Mallios and E Zafiris ldquoThe homological Kahler-de Rhamdifferential mechanism part I application in general theory ofrelativityrdquo Advances in Mathematical Physics vol 2011 ArticleID 191083 14 pages 2011

[26] A Mallios and E Zafiris ldquoThe homological Kahler-de Rhamdifferential mechanism II Sheaf-theoretic localization of quan-tum dynamicsrdquo Advances in Mathematical Physics vol 2011Article ID 189801 13 pages 2011

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Stochastic AnalysisInternational Journal of


introduced previously by Aharonov and Bohm [3 4] see also[5] The conceptual precursors of this astonishing discoverywhich has been unnoticed in the foundations of quantumtheory for more than 60 years are the work of Pancharatnamin polarization optics and the Aharonov-Bohm effect inelectromagnetism In 1956 Pancharatnam [6] realized thatin order to understand interference phenomena it is notrequired to know the absolute phase but only the relativephase difference between light beams in different states ofpolarization For two light beams this relative phase is givenby the phase argument of their complex-valued scalar innerproduct In 1959 Aharonov and Bohm showed that thereexists a measurable global phase shift in the interferencepattern of two coherent wave functions of a charged particledue to the existence of an electromagnetic field constrainedwithin an infinitely long solenoid even in case that the wavefunctions are localized within regions outside the solenoidwhere the strength of the electromagnetic field vanishes

The general Berry-generation mechanism of an exper-imentally observable global phase factor which is of ageometric or topological origin can be described briefly asfollows A quantum system undergoing a slowly evolving(adiabatic) cyclic evolution retains a trace of its motionafter coming back to its original physical state This trace isexpressed by means of a complex phase factor in the wavefunction (state vector) of the system called Berryrsquos phase orthe geometric phase The cyclic evolution which is better tobe thought of as a periodicity property of the state vector ofa quantum system is driven by a Hamiltonian bearing animplicit time dependence through a set of control variablesFor instance we may think of external electric or magneticfields which define the Hamiltonian parametric dependenceof a charged particle The adiabatic condition defines aconstraint of parallel transport specified by the requirementthat the implicit time dependence of the Hamiltonian issufficiently slow so that the state vector stays in the eigenspaceof the same instantaneous eigenvalue of the HamiltonianIntuitively once the state vector is prepared in an instan-taneous eigenstate of the Hamiltonian with an eigenvaluewhich is separated from the neighboring eigenstates by afinite energy gap then it remains there during its transportwithin a finite temporal period We may think of the spaceof control variables as a slowly varying environment withrespect to which a state vector (eigenvector of the Hamil-tonian localized at the corresponding eigenspace) displays ahistory dependent geometric effect When the environmentreturns to its original state the system also does but foran additional global topological or geometric phase factorDue to the implicit temporal dependence imposed by thetime parameterization of a closed path in the environmentalparameters of the control space this global geometric phasefactor is thought of as a trace of the motion encoding theglobal geometric or topological features of the control spaceThe Berry phase is a complex number of modulus one and isexperimentally observable The two most important featuresregarding the experimental detection of a quantum globalphase are as follows (i) it is a statistical object and (ii) it canbe measured only relatively Thus it becomes observable bycomparing the historical evolution of two distinct statistical

ensembles of systems through their interference pattern TheBerry phase is geometric or topological because it dependssolely on the geometry or topology of the control spacepathway along which the state vector is transported It doesnot depend neither on the temporal metric duration of theevolution nor on the particular dynamics that is applied tothe system

It is instructive to distinguish briefly between geometricand topological global phase factors as follows In the case of asimply connected base space of control variables for examplea sphere the global observable phase factor is of a geometricorigin More precisely it appears due to the curvature of thenonintegrable connection via which the transport of statevectors takes place In the case of a nonsimply connectedspace of control variables a global observable phase factorof a topological origin is always going to appear even if theconnection is integrable The integrability property meansthat the connection does not depend on the closed pathtraced on the control space under the proviso that theseclosed paths belong to the same homotopy equivalence classIntuitively speaking a closed path is homotopically nontrivialif it encloses a hole (eg an inaccessible region) so that itcannot be contracted continuously to a point in the controlspace Depending on the topological connectivity propertiesof this space closed paths can be classified homotopically bymeans of global invariants for example the winding numberof a closed path

The particular significance of the concepts of relativetopological and geometric phase factors from the viewpointof our theoretical scheme is that they mark a distinctivepoint in the history of quantum theory where for the firsttime the significance of global observables as distinct entitiesfrom local observables is realized and made explicit throughprecise physical models which have found concrete experi-mental applications In particular these global observablesfor example global relative phases are obtained via anintegration procedure of some differential gauge potentialover a contour represented as a closed path or loop ona base space of control variables (or localization space)which is implicitly parameterized continuously by an externaltemporal parameter This is usually referred to as a cyclicevolution of a quantum state over a space of parameterswhich is implicitly a continuous or smooth function of time Itis also assumed that this evolution is driven by a Hamiltonianwhich is expressed as a function of the localization spaceparameters bearing an implicit time dependence by consid-ering variation of the temporal parameter Intuitively loopsin the space of control variables have the potential to probethe geometrically or topologically invariant information ofthe localization space for example by encircling a hole or aninaccessible region This nontrivial geometric or topologicalinformation of global significance is measured in terms ofglobal holonomy phase factors via the procedure of liftingclosed paths from the localization base space to the statesor observables defined over it according to some paralleltransport constraint (like the adiabatic one) technicallycalled a connection Due to the implicit time dependenceparameterizing this procedure if we trace continuously a loopon the base space then this loop can be lifted to the implicitly








119909of



E|119880cong

119898

⨁(A|119880) = (A|

119880)119898

(1)






E|119880cong A|119880 (2)

where A|119880




of119883



E|119880cong C|119880

(3)



120597120595

120597119905= (120577

119894(119905)) 120595 (4)



(120577119894) 120595119899(120577119894) = 119864119899(120577119894) 120595119899(120577119894) (5)


terized eigenvalue




120578120572 E|119880120572

cong (A|119880120572

)119898

120578120573 E|119880120573

cong (A|119880120573

)119898

(6)


120578120572(119909) E

119909cong A119898119909

(7)


120572cap119880120573


119892120572120573(119909) = 120578

120572(119909)minus1

∘ 120578120573(119909) A119898

119909cong A119898119909 (8)


Consequently 119892120572120573



120572120573∘

119892120573120574= 119892120572120574





Llarrrarr (119892120572120573) isin 1198851

(U A) (9)




120572) in the set


1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (10)


Δ0

1198620

(U A) 997888rarr 1198621

(U A) (11)



120572) in 1198611(U A)

Δ0

(119905minus1

120572) = 119905120572sdot 119905minus1

120573 (12)





1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572) (13)




119868119904119900 (L) (119883) cong 1198671 (119883 A) (14)




LotimesALlowast







984858 (120572119904 120573119905) = 120572 sdot 120573 sdot 984858 (119904 119905)(17)




120572120573in 1198851(U A)






1-cocycle 119892120572120573



120572120573in 1198851(US1)



0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (19)



0 997888rarr Z120580



(20)


0 997888rarr Z120580








(119883Z) 997888rarr 1198671

(119883A) 997888rarr 1198671

(119883 A)

997888rarr 1198672

(119883Z) 997888rarr 1198672

(119883A) 997888rarr 1198672

(119883 A)


(22)


1198671

(119883A) = 1198672 (119883A) = 0 (23)


0 997888rarr 1198671

(119883 A)120575119888

997888997888997888rarr 1198672

(119883Z) 997888rarr 0 (24)


120575119888 1198671

(119883 A) cong997888997888997888rarr 1198672

(119883Z) (25)


Iso (L) (119883) cong 1198672 (119883Z) (26)









(A) otimesAE (27)



(119886) (28)


nablaE = 1198890

+ 120596 (29)

where 120596 = 120596120572120573




1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (30)


119890119880

equiv 119880 (119890120572)1le120572le119899

(31)


119904 =

119899

sum120572=1

119904120572119890120572

(32)



nablaE (119904) =119899

sum120572=1

(119904120572nablaE (119890120572) + 119890120572 otimes 119889

0

(119904120572))

nablaE (119890120572) =119899

sum120572=1

119890120572otimes 120596120572120573 1 le 120572 120573 le 119899

(33)


where 120596 = 120596120572120573


nablaE (119904) =119899

sum120572=1

119890120572otimes (1198890

(119904120572) +

119899

sum120573=1

119904120573120596120572120573)

equiv (1198890

+ 120596) (119904)

(34)


nablaE = 1198890

+ 120596 (35)



119880 119890120572=1119899

and ℎ119881 equiv 119881 ℎ120573=1119899



ℎ120573=

119899

sum120572=1

119892120572120573119890120572 (36)



1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (37)


LotimesALlowast






1198890

984858 (119904 119905) = 984858 (nabla119904 119905) + 984858 (119904 nabla119905) (39)




984858 Ω (L) oplus L 997888rarr Ω(L)

984858 (119904 otimes 119905 1199041015840

) = 984858 (119904 1199041015840

) sdot 119905(40)





120596 + 120596 = 1198890

(984858) (41)


1198890

A 997888rarr Ω1 (42)


1198890

(119906) = 119906minus1

sdot 1198890

(119906) (43)


Ker (1198890) = C (44)


1198890

(984858) = 0 (45)







(U A) times 1198620 (U Ω1) (46)



1205750

(120596120572) = 1198890

(119892120572120573) (47)


120572120573in 1198851(U A)


120572with respect




(U A) times 1198620 (U Ω1) (48)


1198620


120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)











1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)



(120596120572) = 120596120573minus 120596120572 (52)




(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1



1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)





1-cocycle 119892120572120573





120572) isin




nabla1015840


(55)


984858)



nabla1015840

= ℎnablaℎminus1

(56)








nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119909of



E|119880cong

119898

⨁(A|119880) = (A|

119880)119898

(1)






E|119880cong A|119880 (2)

where A|119880




of119883



E|119880cong C|119880

(3)



120597120595

120597119905= (120577

119894(119905)) 120595 (4)



(120577119894) 120595119899(120577119894) = 119864119899(120577119894) 120595119899(120577119894) (5)


terized eigenvalue




120578120572 E|119880120572

cong (A|119880120572

)119898

120578120573 E|119880120573

cong (A|119880120573

)119898

(6)


120578120572(119909) E

119909cong A119898119909

(7)


120572cap119880120573


119892120572120573(119909) = 120578

120572(119909)minus1

∘ 120578120573(119909) A119898

119909cong A119898119909 (8)


Consequently 119892120572120573



120572120573∘

119892120573120574= 119892120572120574





Llarrrarr (119892120572120573) isin 1198851

(U A) (9)




120572) in the set


1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (10)


Δ0

1198620

(U A) 997888rarr 1198621

(U A) (11)



120572) in 1198611(U A)

Δ0

(119905minus1

120572) = 119905120572sdot 119905minus1

120573 (12)





1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572) (13)




119868119904119900 (L) (119883) cong 1198671 (119883 A) (14)




LotimesALlowast







984858 (120572119904 120573119905) = 120572 sdot 120573 sdot 984858 (119904 119905)(17)




120572120573in 1198851(U A)






1-cocycle 119892120572120573



120572120573in 1198851(US1)



0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (19)



0 997888rarr Z120580



(20)


0 997888rarr Z120580








(119883Z) 997888rarr 1198671

(119883A) 997888rarr 1198671

(119883 A)

997888rarr 1198672

(119883Z) 997888rarr 1198672

(119883A) 997888rarr 1198672

(119883 A)


(22)


1198671

(119883A) = 1198672 (119883A) = 0 (23)


0 997888rarr 1198671

(119883 A)120575119888

997888997888997888rarr 1198672

(119883Z) 997888rarr 0 (24)


120575119888 1198671

(119883 A) cong997888997888997888rarr 1198672

(119883Z) (25)


Iso (L) (119883) cong 1198672 (119883Z) (26)









(A) otimesAE (27)



(119886) (28)


nablaE = 1198890

+ 120596 (29)

where 120596 = 120596120572120573




1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (30)


119890119880

equiv 119880 (119890120572)1le120572le119899

(31)


119904 =

119899

sum120572=1

119904120572119890120572

(32)



nablaE (119904) =119899

sum120572=1

(119904120572nablaE (119890120572) + 119890120572 otimes 119889

0

(119904120572))

nablaE (119890120572) =119899

sum120572=1

119890120572otimes 120596120572120573 1 le 120572 120573 le 119899

(33)


where 120596 = 120596120572120573


nablaE (119904) =119899

sum120572=1

119890120572otimes (1198890

(119904120572) +

119899

sum120573=1

119904120573120596120572120573)

equiv (1198890

+ 120596) (119904)

(34)


nablaE = 1198890

+ 120596 (35)



119880 119890120572=1119899

and ℎ119881 equiv 119881 ℎ120573=1119899



ℎ120573=

119899

sum120572=1

119892120572120573119890120572 (36)



1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (37)


LotimesALlowast






1198890

984858 (119904 119905) = 984858 (nabla119904 119905) + 984858 (119904 nabla119905) (39)




984858 Ω (L) oplus L 997888rarr Ω(L)

984858 (119904 otimes 119905 1199041015840

) = 984858 (119904 1199041015840

) sdot 119905(40)





120596 + 120596 = 1198890

(984858) (41)


1198890

A 997888rarr Ω1 (42)


1198890

(119906) = 119906minus1

sdot 1198890

(119906) (43)


Ker (1198890) = C (44)


1198890

(984858) = 0 (45)







(U A) times 1198620 (U Ω1) (46)



1205750

(120596120572) = 1198890

(119892120572120573) (47)


120572120573in 1198851(U A)


120572with respect




(U A) times 1198620 (U Ω1) (48)


1198620


120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)











1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)



(120596120572) = 120596120573minus 120596120572 (52)




(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1



1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)





1-cocycle 119892120572120573





120572) isin




nabla1015840


(55)


984858)



nabla1015840

= ℎnablaℎminus1

(56)








nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E|119880cong C|119880

(3)



120597120595

120597119905= (120577

119894(119905)) 120595 (4)



(120577119894) 120595119899(120577119894) = 119864119899(120577119894) 120595119899(120577119894) (5)


terized eigenvalue




120578120572 E|119880120572

cong (A|119880120572

)119898

120578120573 E|119880120573

cong (A|119880120573

)119898

(6)


120578120572(119909) E

119909cong A119898119909

(7)


120572cap119880120573


119892120572120573(119909) = 120578

120572(119909)minus1

∘ 120578120573(119909) A119898

119909cong A119898119909 (8)


Consequently 119892120572120573



120572120573∘

119892120573120574= 119892120572120574





Llarrrarr (119892120572120573) isin 1198851

(U A) (9)




120572) in the set


1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (10)


Δ0

1198620

(U A) 997888rarr 1198621

(U A) (11)



120572) in 1198611(U A)

Δ0

(119905minus1

120572) = 119905120572sdot 119905minus1

120573 (12)





1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572) (13)




119868119904119900 (L) (119883) cong 1198671 (119883 A) (14)




LotimesALlowast







984858 (120572119904 120573119905) = 120572 sdot 120573 sdot 984858 (119904 119905)(17)




120572120573in 1198851(U A)






1-cocycle 119892120572120573



120572120573in 1198851(US1)



0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (19)



0 997888rarr Z120580



(20)


0 997888rarr Z120580








(119883Z) 997888rarr 1198671

(119883A) 997888rarr 1198671

(119883 A)

997888rarr 1198672

(119883Z) 997888rarr 1198672

(119883A) 997888rarr 1198672

(119883 A)


(22)


1198671

(119883A) = 1198672 (119883A) = 0 (23)


0 997888rarr 1198671

(119883 A)120575119888

997888997888997888rarr 1198672

(119883Z) 997888rarr 0 (24)


120575119888 1198671

(119883 A) cong997888997888997888rarr 1198672

(119883Z) (25)


Iso (L) (119883) cong 1198672 (119883Z) (26)









(A) otimesAE (27)



(119886) (28)


nablaE = 1198890

+ 120596 (29)

where 120596 = 120596120572120573




1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (30)


119890119880

equiv 119880 (119890120572)1le120572le119899

(31)


119904 =

119899

sum120572=1

119904120572119890120572

(32)



nablaE (119904) =119899

sum120572=1

(119904120572nablaE (119890120572) + 119890120572 otimes 119889

0

(119904120572))

nablaE (119890120572) =119899

sum120572=1

119890120572otimes 120596120572120573 1 le 120572 120573 le 119899

(33)


where 120596 = 120596120572120573


nablaE (119904) =119899

sum120572=1

119890120572otimes (1198890

(119904120572) +

119899

sum120573=1

119904120573120596120572120573)

equiv (1198890

+ 120596) (119904)

(34)


nablaE = 1198890

+ 120596 (35)



119880 119890120572=1119899

and ℎ119881 equiv 119881 ℎ120573=1119899



ℎ120573=

119899

sum120572=1

119892120572120573119890120572 (36)



1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (37)


LotimesALlowast






1198890

984858 (119904 119905) = 984858 (nabla119904 119905) + 984858 (119904 nabla119905) (39)




984858 Ω (L) oplus L 997888rarr Ω(L)

984858 (119904 otimes 119905 1199041015840

) = 984858 (119904 1199041015840

) sdot 119905(40)





120596 + 120596 = 1198890

(984858) (41)


1198890

A 997888rarr Ω1 (42)


1198890

(119906) = 119906minus1

sdot 1198890

(119906) (43)


Ker (1198890) = C (44)


1198890

(984858) = 0 (45)







(U A) times 1198620 (U Ω1) (46)



1205750

(120596120572) = 1198890

(119892120572120573) (47)


120572120573in 1198851(U A)


120572with respect




(U A) times 1198620 (U Ω1) (48)


1198620


120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)











1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)



(120596120572) = 120596120573minus 120596120572 (52)




(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1



1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)





1-cocycle 119892120572120573





120572) isin




nabla1015840


(55)


984858)



nabla1015840

= ℎnablaℎminus1

(56)








nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572) in 1198611(U A)

Δ0

(119905minus1

120572) = 119905120572sdot 119905minus1

120573 (12)





1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572) (13)




119868119904119900 (L) (119883) cong 1198671 (119883 A) (14)




LotimesALlowast







984858 (120572119904 120573119905) = 120572 sdot 120573 sdot 984858 (119904 119905)(17)




120572120573in 1198851(U A)






1-cocycle 119892120572120573



120572120573in 1198851(US1)



0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (19)



0 997888rarr Z120580



(20)


0 997888rarr Z120580








(119883Z) 997888rarr 1198671

(119883A) 997888rarr 1198671

(119883 A)

997888rarr 1198672

(119883Z) 997888rarr 1198672

(119883A) 997888rarr 1198672

(119883 A)


(22)


1198671

(119883A) = 1198672 (119883A) = 0 (23)


0 997888rarr 1198671

(119883 A)120575119888

997888997888997888rarr 1198672

(119883Z) 997888rarr 0 (24)


120575119888 1198671

(119883 A) cong997888997888997888rarr 1198672

(119883Z) (25)


Iso (L) (119883) cong 1198672 (119883Z) (26)









(A) otimesAE (27)



(119886) (28)


nablaE = 1198890

+ 120596 (29)

where 120596 = 120596120572120573




1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (30)


119890119880

equiv 119880 (119890120572)1le120572le119899

(31)


119904 =

119899

sum120572=1

119904120572119890120572

(32)



nablaE (119904) =119899

sum120572=1

(119904120572nablaE (119890120572) + 119890120572 otimes 119889

0

(119904120572))

nablaE (119890120572) =119899

sum120572=1

119890120572otimes 120596120572120573 1 le 120572 120573 le 119899

(33)


where 120596 = 120596120572120573


nablaE (119904) =119899

sum120572=1

119890120572otimes (1198890

(119904120572) +

119899

sum120573=1

119904120573120596120572120573)

equiv (1198890

+ 120596) (119904)

(34)


nablaE = 1198890

+ 120596 (35)



119880 119890120572=1119899

and ℎ119881 equiv 119881 ℎ120573=1119899



ℎ120573=

119899

sum120572=1

119892120572120573119890120572 (36)



1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (37)


LotimesALlowast






1198890

984858 (119904 119905) = 984858 (nabla119904 119905) + 984858 (119904 nabla119905) (39)




984858 Ω (L) oplus L 997888rarr Ω(L)

984858 (119904 otimes 119905 1199041015840

) = 984858 (119904 1199041015840

) sdot 119905(40)





120596 + 120596 = 1198890

(984858) (41)


1198890

A 997888rarr Ω1 (42)


1198890

(119906) = 119906minus1

sdot 1198890

(119906) (43)


Ker (1198890) = C (44)


1198890

(984858) = 0 (45)







(U A) times 1198620 (U Ω1) (46)



1205750

(120596120572) = 1198890

(119892120572120573) (47)


120572120573in 1198851(U A)


120572with respect




(U A) times 1198620 (U Ω1) (48)


1198620


120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)











1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)



(120596120572) = 120596120573minus 120596120572 (52)




(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1



1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)





1-cocycle 119892120572120573





120572) isin




nabla1015840


(55)


984858)



nabla1015840

= ℎnablaℎminus1

(56)








nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(119883Z) 997888rarr 1198671

(119883A) 997888rarr 1198671

(119883 A)

997888rarr 1198672

(119883Z) 997888rarr 1198672

(119883A) 997888rarr 1198672

(119883 A)


(22)


1198671

(119883A) = 1198672 (119883A) = 0 (23)


0 997888rarr 1198671

(119883 A)120575119888

997888997888997888rarr 1198672

(119883Z) 997888rarr 0 (24)


120575119888 1198671

(119883 A) cong997888997888997888rarr 1198672

(119883Z) (25)


Iso (L) (119883) cong 1198672 (119883Z) (26)









(A) otimesAE (27)



(119886) (28)


nablaE = 1198890

+ 120596 (29)

where 120596 = 120596120572120573




1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (30)


119890119880

equiv 119880 (119890120572)1le120572le119899

(31)


119904 =

119899

sum120572=1

119904120572119890120572

(32)



nablaE (119904) =119899

sum120572=1

(119904120572nablaE (119890120572) + 119890120572 otimes 119889

0

(119904120572))

nablaE (119890120572) =119899

sum120572=1

119890120572otimes 120596120572120573 1 le 120572 120573 le 119899

(33)


where 120596 = 120596120572120573


nablaE (119904) =119899

sum120572=1

119890120572otimes (1198890

(119904120572) +

119899

sum120573=1

119904120573120596120572120573)

equiv (1198890

+ 120596) (119904)

(34)


nablaE = 1198890

+ 120596 (35)



119880 119890120572=1119899

and ℎ119881 equiv 119881 ℎ120573=1119899



ℎ120573=

119899

sum120572=1

119892120572120573119890120572 (36)



1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (37)


LotimesALlowast






1198890

984858 (119904 119905) = 984858 (nabla119904 119905) + 984858 (119904 nabla119905) (39)




984858 Ω (L) oplus L 997888rarr Ω(L)

984858 (119904 otimes 119905 1199041015840

) = 984858 (119904 1199041015840

) sdot 119905(40)





120596 + 120596 = 1198890

(984858) (41)


1198890

A 997888rarr Ω1 (42)


1198890

(119906) = 119906minus1

sdot 1198890

(119906) (43)


Ker (1198890) = C (44)


1198890

(984858) = 0 (45)







(U A) times 1198620 (U Ω1) (46)



1205750

(120596120572) = 1198890

(119892120572120573) (47)


120572120573in 1198851(U A)


120572with respect




(U A) times 1198620 (U Ω1) (48)


1198620


120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)











1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)



(120596120572) = 120596120573minus 120596120572 (52)




(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1



1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)





1-cocycle 119892120572120573





120572) isin




nabla1015840


(55)


984858)



nabla1015840

= ℎnablaℎminus1

(56)








nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 120596 = 120596120572120573


nablaE (119904) =119899

sum120572=1

119890120572otimes (1198890

(119904120572) +

119899

sum120573=1

119904120573120596120572120573)

equiv (1198890

+ 120596) (119904)

(34)


nablaE = 1198890

+ 120596 (35)



119880 119890120572=1119899

and ℎ119881 equiv 119881 ℎ120573=1119899



ℎ120573=

119899

sum120572=1

119892120572120573119890120572 (36)



1205961015840

= 119892minus1

120596119892 + 119892minus1

1198890

119892 (37)


LotimesALlowast






1198890

984858 (119904 119905) = 984858 (nabla119904 119905) + 984858 (119904 nabla119905) (39)




984858 Ω (L) oplus L 997888rarr Ω(L)

984858 (119904 otimes 119905 1199041015840

) = 984858 (119904 1199041015840

) sdot 119905(40)





120596 + 120596 = 1198890

(984858) (41)


1198890

A 997888rarr Ω1 (42)


1198890

(119906) = 119906minus1

sdot 1198890

(119906) (43)


Ker (1198890) = C (44)


1198890

(984858) = 0 (45)







(U A) times 1198620 (U Ω1) (46)



1205750

(120596120572) = 1198890

(119892120572120573) (47)


120572120573in 1198851(U A)


120572with respect




(U A) times 1198620 (U Ω1) (48)


1198620


120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)











1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)



(120596120572) = 120596120573minus 120596120572 (52)




(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1



1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)





1-cocycle 119892120572120573





120572) isin




nabla1015840


(55)


984858)



nabla1015840

= ℎnablaℎminus1

(56)








nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(U A) times 1198620 (U Ω1) (46)



1205750

(120596120572) = 1198890

(119892120572120573) (47)


120572120573in 1198851(U A)


120572with respect




(U A) times 1198620 (U Ω1) (48)


1198620


120596120573= 119892minus1

120572120573120596120572119892120572120573+ 119892minus1

1205721205731198890

119892120572120573

120596120573= 120596120572+ 119892minus1

1205721205731198890

119892120572120573

(49)











1205750

(120596120572) = 1198890

(119892120572120573) (50)

where1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(51)



(120596120572) = 120596120573minus 120596120572 (52)




(U A) times 1198620 (U Ω1) (53)

where |119892120572120573| = 1



1205750

(120596120572) = 1198890

(119892120572120573)

120596 + 120596 = 1198890

(984858)

(54)





1-cocycle 119892120572120573





120572) isin




nabla1015840


(55)


984858)



nabla1015840

= ℎnablaℎminus1

(56)








nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












nabla1015840

) larrrarr (119892120572120573sdot 1198921015840

120572120573 120596120572+ 1205961015840

120572) (57)


1198890

(119892120572120573sdot 1198921015840

120572120573) = 1198890

(119892120572120573) + 1198890

(1198921015840

120572120573)

= 1205750

(120596120572) + 1205750

(1205961015840

120572) = 1205750

(120596120572+ 1205961015840

120572)

(58)





120572120573| = 1 the set of gauge




120572120573 120596120572) and (1198921015840

120572120573 1205961015840120572) determine gauge



119886) isin


1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

119886)

1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886)

(59)


120572120573= 120601120572∘ 120601minus1120573isin 1198851(U A) and 1198921015840

120572120573= 120595120572∘ 120595minus1120573isin


119808|U120572

119819998400|U120572

119808|U120572

119819|U120572

120601120572

h120572

t120572

120595120572 (60)





1198921015840

120572120573= 119905120572sdot 119892120572120573sdot 119905minus1

120573 (61)

that is 1198921015840120572120573

is similar to 119892120572120573




Δ0

1198620

(U A) 997888rarr 1198621

(U A)

1198921015840

120572120573= Δ0

(119905minus1

120572) sdot 119892120572120573

1198921015840

120572120573sdot 119892minus1

120572120573= Δ0

(119905minus1

120572)

(62)


119886) isin 1198620(U A) =




nabla1015840




1205961015840

120572= 119905120572120596120572119905minus1

120572+ 1199051205721198890

(119905minus1

120572)

1205961015840

120572= 120596120572+ 1198890

(119905minus1

120572)

(63)


1205961015840

120572minus 120596120572= 1198890

(119905minus1

119886) (64)





(A)




A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











A 997888rarr Ω1





E 997888rarr Ω1



or equivalently

E 997888rarr Ω1



where the morphism

nabla119899

Ω119899


(A) otimesAE (69)


nabla119899




nabla1

∘ nabla0


2

(A) otimesAE = Ω2

(E) (71)







E 997888rarr Ω1




Rnabla= 0 (73)











isin





Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (74)


1198892 Rnabla

1003816100381610038161003816119880 = Rnabla


1003816100381610038161003816119880 (75)



120572120573


Rnabla

119892


(Rnabla) 119892 (76)




Rnabla

1003816100381610038161003816119880 = 1198891

120596 + 120596 and 120596 (77)



Rnabla

119892


(Rnabla) 119892 (78)


LotimesALlowast





Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Rnabla

119892


120596 (80)



1198892Rnabla= 0 (81)







1198891

∘ 1198890

= 0 (82)


1198891

∘ 1198890

= 0 (83)

we obtain

1198891

(1205961015840

120572) = 1198891

(120596120572) (84)


1198892

∘ 1198891

120596120572= 1198892R = 0 (85)










119888 namely over









1198672

(119883Z) 997888rarr 1198672

(119883C) (87)


0 997888rarr Z120580



Ker (exp) = Im (120580) cong Z (89)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (90)


Ker (1198890) = Im (120576) cong C (91)



119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119808

119808

d0119808

id

exp

1

2120587id0

d0

(92)


2120587119894 sdot 1198890

= 1198890

∘ exp (93)


H1(X A) H

2(XZ)

H1(X d

0119808) H

2(XC)

1

2120587id0

120580lowast

120575c

cong

(94)


120572120573



0 997888rarr Z120580


119892120572120573= exp (119908

120572120573)

(95)


120575119888(119908120572120573) = (119911

120572120573120574) isin 1198852

(UZ) (96)


120572120573) so that

120575119888(119908120572120573) = (119911

120572120573120574)

=1

2120587119894(ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UZ)

(97)


R = (119889120596120572) isin 1198850

(U 119889Ω1

) (98)


1205750

(120596120572) = 1198890

(119892120572120573) (99)

where

1198890

(119892120572120573) = 119892minus1

120572120573sdot 1198890

119892120572120573

(100)



1205750

(120596120572) = 120596120573minus 120596120572 (101)

Thus we obtain

1205750

(120596120572) = 2120587119894 sdot 119889

0

(119908120572120573) (102)


1

2120587119894(120596120573minus 120596120572) = 1198890

(ln (119892120572120573)) (103)


0 997888rarr C120576

997888997888rarr A 1198890

997888997888rarr 1198890A 997888rarr 0 (104)


120575 (ln (119892120572120573)) = (ln (119892

120572120573) + ln (119892

120573120574) minus ln (119892

120572120574))

isin 1198852

(UC)

(105)


[R] = 2120587119894 sdot [(119911120572120573120574)] isin 119867

2

(119883C)

[(119892120572120573)] =

1

2120587119894[R] = [(119911

120572120573120574)] isin 119867

2

(119883Z)

[R] isin Im (1198672

(119883Z) 997888rarr 1198672

(119883C))

(106)


119888 which are







Iso (L nabla) = sum

RisinΩ2119888Z




984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











984858)












S1

times A (119880) 997888rarr A (119880)

(120585 119891) 997891997888rarr 120585 sdot 119891(108)


1198851

(US1

) times 1198851

(U A) 997888rarr 1198851

(U A)

(120585120572120573) sdot (119892120572120573) = (120585

120572120573sdot 119892120572120573)

(109)

where (120585120572120573) isin 1198851(US1) and (119892

120572120573) isin 1198851(U A) This free


1198671

(119883S1

) otimes 1198671

(119883 A) 997888rarr 1198671

(119883 A)

[(120585120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)]

(110)

where [(120585120572120573)] isin 1198671(119883S1) and [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)






120572120573sdot 119892120572120573)

(111)



120575 (120596120572) = 1198890

(120585120572120573sdot 119892120572120573) (112)

Now given that

Ker (1198890) = C (113)



120572120573)] isin 1198671(119883S1) and



120572A(119880120572) such that

120585120572120573= Δ0

(119905minus1

120572) (114)


[(120585120572120573)] = 1 isin 119867

1

(119883 A) (115)




1205961015840

120572= 120596120572

(116)

and thus

1198890

(119905minus1

119886) = minus119889

0

(119905119886) = 0 (117)



119886isin Ker(1198890) =


119886) isin 119862

0


[(120585120572120573)] = 1 isin 119867

1

(119883S1

) (118)



120572120573)] = 1 isin 1198671(119883 C) where (120585

120572120573) isin 1198851(U C)









the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















the pairing

1198671(119883) times 119867

1

(119883119880 (1)) 997888rarr 119880 (1) (119)





Hom (1205871(119883) C) cong 119867

1

(119883 C) (120)


Hom (1205871(119883) S

1

) cong 1198671

(119883S1

) (121)






1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (122)


Ker (1198890) = C (123)


120572ofΩ1 for


Im (1198890

) = Ker (1198891) (124)




1 997888rarr C120598

997888997888rarr A1198890

997888997888997888rarr Ω11198891

997888997888997888rarr 1198891

Ω1

997888rarr 0 (125)



120572) of


(120579120572) isin 1198620

(UKer (1198891)) = (120579120572) isin 1198620

(U Im (1198890

))

= 1198890

(1198620

(U A)) (126)




120579120572= 1198890

(119905minus1

120572) (127)



120577120572120573= 119905minus1

120573119905120572 (128)


1205750

(120579120572) = 120579120573minus 120579120572= 1198890

(119905minus1

120573) minus 1198890

(119905minus1

120572) = 1198890

(119905minus1

120573119905120572)

= 1198890

(120577120572120573)

(129)






Im (1198890

) = Ker (1198891) (130)


984858) (L1015840 nabla1015840

984858)


R = (119889120596120572) = (119889120596

1015840

120572)

119889 (120596120572minus 1205961015840

120572) = 0

(131)






1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1-forms

(120596120572minus 1205961015840

120572) isin 1198620

(UKer (1198891))

= (120596120572minus 1205961015840

120572) isin 1198620

(U Im (1198890

)) = 1198890

(1198620

(U A)) (132)



120596120572minus 1205961015840

120572= 1198890

(119905minus1

120572)

120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(133)



follows

11989210158401015840

120572120573= 120577120572120573sdot 1198921015840

120572120573= 119905120572sdot 1198921015840

120572120573sdot 119905minus1

120573

12059610158401015840

120572= 120596120572= 1205961015840

120572+ 1198890

(119905minus1

120572)

(134)

such that

1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) (135)




984858) (L1015840 nabla1015840




984858)



1198890

(11989210158401015840

120572120573) = 1205750

(12059610158401015840

120572) = 1205750

(120596120572) = 1198890

(119892120572120573) (136)

We conclude that

1198890

(11989210158401015840

120572120573) minus 1198890

(119892120572120573) = 0

1198890

(11989210158401015840

120572120573sdot 119892minus1

120572120573) = 0

(137)


Poincare Lemma

(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(U C) (138)


(11989210158401015840

120572120573sdot 119892minus1

120572120573) isin 1198851

(US1

) (139)


11989210158401015840

120572120573= 120585120572120573sdot 119892120572120573 (140)

where 120585120572120573


[11989210158401015840

120572120573] = [(120585

120572120573)] sdot [(119892

120572120573)] = [(120585

120572120573sdot 119892120572120573)] (141)

where [(120585120572120573)] isin 1198671(119883S1) [(119892

120572120573)] isin 1198671(119883 A) cong Iso(L)


[(L1015840 nabla1015840984858)] = [(L10158401015840 nabla

984858)] = [(120585 sdot L nabla)]

= [(120585)] sdot [(L nabla)] (142)
















Acknowledgment




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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