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Classical GeometryClassical Geometryofof
Quantum Quantum Integrabi l i tyIntegrabi l i tyandand
Gauge TheoryGauge TheoryNikita Nekrasov
IHES
This is a work onexperimental
theoretical physics•
In collaboration with
Alexei Rosly(ITEP)
and
Samson Shatashvili(HMI and Trinity College Dublin)
Based on
NN, S.Shatashvili,arXiv:0901.4744, arXiv:0901.4748, arXiv: 0908.4052
NN, E.WittenarXiv:1002.0888
Earlier workG.Moore, NN, S.Shatashvili.,arXiv:hep-th/9712241;A.Gerasimov, S.Shatashvili.arXiv:0711.1472, arXiv:hep-th/0609024
Earlier work on instanton calculusA.Losev, NN, S.Shatashvili.,arXiv:hep-th/9711108, arXiv:hep-th/9911099;NN arXiv:hep-th/0206161;The papers ofN.Dorey, T.Hollowood , V.Khoze, M.Mattis,Earlier work on separated variables and D-branesA.Gorsky, NN, V.Roubtsov, arXiv:hep-th/9901089;
The supersymmetricgauge theories
(including exotics)With as little as 4 supersymmetries
on the one hand
The more precise slogan is:The supersymmetric vacua of
the (finite volume) gauge theory
are the eigenstates of aquantum integrable system
The correspondenceThe supersymmetric vacua of
gauge theory
are the eigenstates of aquantum integrable system
The correspondenceThe supersymmetric vacua of
gauge theory
are the eigenstates of aquantum integrable system
OperatorsThe « twisted chiral ring » operators
O1, O2, O3, … , On
map to quantum Hamiltonians
Η1, Η2, Η3, … , Ηn
EigenvaluesThe vacuum expectation values ofthe twisted chiral ring operators
map to the energy and other eigenvalues on theintegrable side
There are generalizations forevery spin group,
every finite dimensionalrepresentation,Inhomogeneity,
Anisotropy (XXZ, XYZ)
The main ingredient of thecorrespondence:
The effective twistedsuperpotential of the gauge
theory=
The Yang-Yang function of thequantum integrable system
The effective twistedsuperpotential
Is a multi-valued function on the Coulomb branch of the theory,
depends on the parameters of the theory
The Yang-Yang functionof the quantum integrable
system
The YY function was introducedby C.N.Yang and C.P.Yang in 1969
For the non-linear Schroedinger problem.
The miracle of Bethe ansatz:The spectrum of the quantum
system is described bya classical equation
1
The elliptic Calogero-Mosersystem
N identical particles ona circle of radius β
subject to the two-body interactionelliptic potential
Quantummany-body systems
One is interested in the β-periodic symmetric,L2-normalizable wavefunctions
Many-body systemvs
gauge theory
Theinfinite discrete spectrum
ofthe integrable many-body system
=The vacua of the N=2 d=2 theory
Many-body system:the guess for
the gauge theory
The vacua of the N=2 d=2 theory,Obtained by subjecting the N=2 d=4Theory to an Ω-background in R2
The four dimensional gaugetheory on Σ x R2,
viewed SO(2) equivariantly,can be formally treated as aninfinite dimensional version of atwo dimensional gauge theory
1:56
The cohomological renormalizationgroup applied to the
four dimensional theory
Looks two dimensionalin the infrared:localization at
the « cosmic string »
Bethe equationsFactorized S-matrix
Harish-Chandra, Gindikin-Karpelevich,Olshanetsky-Perelomov, Heckmann,
final result: Opdam
The full superpotentialof N=2* theory leads to the vacuum
equations
q = exp ( - Nβ )
Momentum phase shift
Two-body scattering The finite size corrections
Dictionary
The correspondenceExtends to other integrable sytems:
Toda, relativistic Systems,Perhaps all 1+1 iQFTs
In the mid-nineties of the 20century it was understood,
that
The geometry of the moduli space ofvacua of N=2 supersymmetric gaugetheory is identified with that of a base ofan algebraic integrable system
Donagi, WittenGorsky, Krichever, Marshakov, Mironov, Morozov
The Coulomb branchof the moduli space
of vacua of thed=4 N=2 supersymmetric
gauge theoryis the base
of a complex (algebraic)integrable system
The Coulomb branchof the same theory,
compactified on a circle downto three dimensions
is the phasespace of
the same integrable system
This moduli space is ahyperkahler manifold, and
it can arise both as a Coulombbranch of one susy gauge
theory and as a Higgs branchof another susy theory.
This is the 3d mirrorsymmetry.
In particular, one can startwith a
six dimensional (0,2)ADE
superconformal field theory,and compactify it on
Σ x S1
with the genus g Riemann surface Σ
The resulting effective susygauge theory in three
dimensions will have 8supercharges (with the
appropriate twist along Σ)
The resulting effective susygauge theory in three
dimensions will have theHitchin moduli space as the
moduli space of vacua.The gauge group in Hitchin’sequations will be the group of
the same A,D,E typeas in the definition of the (0,2)
theory.
The mirror theory, for which theHitchin moduli space
is the Coulomb branch,is conjectured Gaiotto, in the A1 case,
to be the SU(2)3g-3
gauge theory in 4d,compactified on a circle,
with some matter hypermultiplets inthe
tri-fundamental and/or adjointrepresentations
One can allow the Riemann surfacewith n punctures,
with some local parametersassociated with the punctures.
The gauge group is then SU(2)3g-3+n
with matter hypermultiplets in thefundamental,
bi-fundamental,tri-fundamental representations,and, sometimes, in the adjoint.
For example,the SU(2) with Nf=4
Corresponds to the Riemann surface of genus zero with 4 punctures.The local data at the puncturesdetermines the masses
For example,the N=2* SU(2) theory
Corresponds to the Riemann surface of genus onewith 1 punctures.The local data at the puncturedetermines the mass of the adjoint
From now on we shall bediscussing these
« generalized quivertheories »
• The integrable system corresponding tothe moduli space of vacua of the 4dtheory is the SU(2) Hitchin system on
The punctured Riemann surface Σ
Hitchin system
Gauge theory on a Riemann surface
The gauge field Aµ and the twistedHiggsfield Φµ in the adjointrepresentation are required to obey:
Hyperkahler structure of MH
• Three complex structures: I,J,K
• Three Kahler forms: ωI , ωJ , ωK
• Three holomorphic symplectic forms:ΩI , ΩJ , ΩK
Hyperkahler structure of MH
Three Kahler forms: ωI , ωJ , ωK
Three holomorphic symplectic forms:
ΩI = ωJ + i ωK,ΩJ = ωK + i ωI,ΩK = ωI + i ωJ
Hyperkahler structure
The linear combinations, parametrized by thepoints on a twistor two-sphere S2
a I + b J + c K, wherea2+b2+c2=1
The integrable structure
In the complex structure I,the holomorphic functions are: for each Beltrami
differential µ(i), i=3g-3+n
The integrable structureThe action-angle variables:
Fix the level of the integrals of motion,ie fix the values of all Hi’s
Equivalently:fix the (spectral)
curve C inside T*ΣDet( λ - Φz ) = 0
Its Jacobian is the Liouville torus, andThe periods of λdz give the special coordinates
ai, aDi
The quantum integrablestructure
The naïve quantization, using thatin the complex structure I
MH is almost = T*MWhere M=BunG
Φz becomes the derivativeHi become the differential operators.
More precisely, one gets the space of twisted (by K1/2M)
differential operators on M=BunG
Now turn on the Ω - deformationOf the Four dimensional gauge theory,
andconclude that the quantum Hitchin
systemIs governed by a Yang-Yang function,
The effective twisted superpotential
Here comes the experimentalfact
The effective twistedsuperpotential,
the YY function of the quantumHitchin system:indeed has a
classical mechanical meaning
MH as themoduli space of GC flat
connectionsIn the complex structure J the
holomorphic variables are:
Aµ = Aµ + i Φµ
which obey (modulo complexified gauge transformations):
F= dA+ [A,A] = 0
MH as themoduli space of GC flat
connectionsIn this complex structure MH
is defined without a reference to thecomplex structure of Σ
MH = Hom ( π1(Σ) , GC )/ GC
MH as themoduli space of GC flat
connectionsHowever MH
Contains interesting complex Lagrangiansubmanifolds which do depend on the
complex structure of Σ
LΣ = the variety of G-opers
Beilinson, DrinfeldDrinfeld, Sokolov
MH as themoduli space of GC flat
connections
Beilinson, DrinfeldDrinfeld, Sokolov
LΣ = the variety of G-opers
The Beltrami differentialµ is fixed, the projectivestructure T is arbitrary,provided
MH as themoduli space of GC flat
connections
LΣ = the variety of G-opers
The Beltrami differential µ is fixed, the projective structure Tis arbitrary, provided it is compatible with the complexstructure defined by
MH as themoduli space of GC flat
connections
LΣ =
The Beltrami differential µ is fixed, the projective structure Tis arbitrary, provided it is compatible with the complexstructure defined by
i.e.
Opers on a sphereFor example, on a two-sphere with n punctures, theseconditions translate to the following definition of the spaceof opers with regular singularities: we are studying thespace of differential operators of second order, of the form
Opers on a sphereAll in all we get a (n-3)-dimensional subvariety in the
2(n-3) dimensional moduli space of flat connections on then-punctured sphere with fixed conjugacy classes of the
monodromies around the punctures:
mi = Tr (gi)
The moduli space is going to be covered by amultitude of Darboux coordinate charts, one per
every pants decomposition
The maximal complex structure degenerations =The weakly coupled gauge theory descriptions of
Gaiotto theories, e.g. for the previous example
The αi coordinates are nothing but the logarithmsof the eigenvalues of the monodromies around
the blue cycles:
The βi coordinates are defined from the local datainvolving the cycle Ci and its
four neighboring cycles(or one, if the blue cycle belongs to a genus one
component) :
The coordinates αi ,βi can bethus explicitly expressed in terms of
the traces of the monodromies:
= Tr (HolC ~ g2g1 )
= Tr gi
B = Tr (HolCv ~ g3g2 )
The theory
Why did the twisted superpotential turn into a generating function?Why did the variety of opers showed up?
What is the meaning of Bethe equations forquantum Hitchin in terms
of this classical symplectic geometry?Why did these hyperbolic coordinates
(which generalize the Fenchel-Nielsen coordinates onTeichmuller space and Goldman coordinates on the moduli
of SU(2) flat connections) become the special coordinates in thetwo dimensional N=2 gauge theory?
What is the relevance of the geometry of hyperbolic polygons forthe M5 brane theory?
For the three dimensional gravity?For the loop quantum gravity?
The theory
Why did the twisted superpotential turn into a generating function,and why did the variety of opers showed up?
This can be understood by viewing the 4d gauge theory as a 2dtheory with an infinite number of fields in two different ways
(NN+EW)