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Two Dimensional Gauge Theories and Quantum Integrable Systems. Nikita Nekrasov IHES Imperial College April 10, 2008. Based on. NN, S.Shatashvili, to appear Prior work: E.Witten, 1992; A.Gorsky, NN; J.Minahan, A.Polychronakos; M.Douglas; ~1993-1994; A.Gerasimov ~1993; - PowerPoint PPT Presentation
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Two Dimensional Gauge Theories
and Quantum Integrable Systems
Two Dimensional Gauge Theories
and Quantum Integrable Systems
Nikita Nekrasov IHES
Imperial College April 10, 2008
Nikita Nekrasov IHES
Imperial College April 10, 2008
Based onBased on
NN, S.Shatashvili, to appear
Prior work:E.Witten, 1992;
A.Gorsky, NN; J.Minahan, A.Polychronakos;M.Douglas; ~1993-1994; A.Gerasimov ~1993;
G.Moore, NN, S.Shatashvili ~1997-1998; A.Losev, NN, S.Shatashvili ~1997-1998;A.Gerasimov, S.Shatashvili ~ 2006-2007
NN, S.Shatashvili, to appear
Prior work:E.Witten, 1992;
A.Gorsky, NN; J.Minahan, A.Polychronakos;M.Douglas; ~1993-1994; A.Gerasimov ~1993;
G.Moore, NN, S.Shatashvili ~1997-1998; A.Losev, NN, S.Shatashvili ~1997-1998;A.Gerasimov, S.Shatashvili ~ 2006-2007
We are going to relate 2,3, and 4 dimensional
susy gauge theorieswith four supersymmetries
N=1 d=4
We are going to relate 2,3, and 4 dimensional
susy gauge theorieswith four supersymmetries
N=1 d=4
And quantum integrable systemssoluble by Bethe Ansatz techniques.
And quantum integrable systemssoluble by Bethe Ansatz techniques.
Mathematically speaking, the cohomology, K-theory and elliptic
cohomology of various gauge theory moduli spaces, like moduli of flat
connections and instantons
Mathematically speaking, the cohomology, K-theory and elliptic
cohomology of various gauge theory moduli spaces, like moduli of flat
connections and instantons
And quantum integrable systemssoluble by Bethe Ansatz techniques.
And quantum integrable systemssoluble by Bethe Ansatz techniques.
For example, we shall relate the XXX Heisenberg magnet
and 2d N=2 SYM theory with some matter
For example, we shall relate the XXX Heisenberg magnet
and 2d N=2 SYM theory with some matter
(pre-)History(pre-)History
In 1992 E.Witten studied two dimensional Yang-Mills theory with the goal to understand the relation
between the physical and topological gravities in 2d.
In 1992 E.Witten studied two dimensional Yang-Mills theory with the goal to understand the relation
between the physical and topological gravities in 2d.
(pre-)History(pre-)History
There are two interesting kinds of Two dimensional Yang-Mills theories There are two interesting kinds of Two dimensional Yang-Mills theories
Yang-Mills theories in 2dYang-Mills theories in 2d(1)
Cohomological YM = twisted N=2 super-Yang-Mills theory,
with gauge group G, whose BPS (or TFT) sector is related to
the intersection theory on the moduli space MG of flat G-connections on
a Riemann surface
(1)
Cohomological YM = twisted N=2 super-Yang-Mills theory,
with gauge group G, whose BPS (or TFT) sector is related to
the intersection theory on the moduli space MG of flat G-connections on
a Riemann surface
Yang-Mills theories in 2dYang-Mills theories in 2d
N=2 super-Yang-Mills theoryN=2 super-Yang-Mills theory
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Yang-Mills theories in 2dYang-Mills theories in 2d
(2) Physical YM =
N=0 Yang-Mills theory, with gauge group G; The moduli space MG of flat G-connections
= minima of the action;The theory is exactly soluble (A.Migdal) with the
help of the Polyakov lattice YM action
(2) Physical YM =
N=0 Yang-Mills theory, with gauge group G; The moduli space MG of flat G-connections
= minima of the action;The theory is exactly soluble (A.Migdal) with the
help of the Polyakov lattice YM action
Yang-Mills theories in 2dYang-Mills theories in 2d
Physical YM Physical YM
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Yang-Mills theories in 2dYang-Mills theories in 2d
Witten found a way to map the BPS sector of the N=2 theory to the N=0 theory.
The result is:
Witten found a way to map the BPS sector of the N=2 theory to the N=0 theory.
The result is:
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Yang-Mills theories in 2dYang-Mills theories in 2d
Two dimensional Yang-Mills partition function is given by the explicit sum
Two dimensional Yang-Mills partition function is given by the explicit sum
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Yang-Mills theories in 2dYang-Mills theories in 2d
In the limit
the partition function computes the volume of MG
In the limit
the partition function computes the volume of MG
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Yang-Mills theories in 2dYang-Mills theories in 2d
Witten’s approach: add twisted superpotential and its conjugate
Witten’s approach: add twisted superpotential and its conjugate
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Yang-Mills theories in 2dYang-Mills theories in 2d
Take a limit Take a limit
In the limit the fields
are infinitely massive and can be integrated out:
one is left with the field content of
the physical YM theory
In the limit the fields
are infinitely massive and can be integrated out:
one is left with the field content of
the physical YM theory
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Yang-Mills theories in 2dYang-Mills theories in 2d
Both physical and cohomological Yang-Millstheories define topological field theories (TFT)
Both physical and cohomological Yang-Millstheories define topological field theories (TFT)
Yang-Mills theories in 2dYang-Mills theories in 2dBoth physical and cohomological Yang-Mills
theories define topological field theories (TFT) Both physical and cohomological Yang-Mills
theories define topological field theories (TFT)
Vacuum states + deformations = quantum mechanics
YM in 2d and particles on a circleYM in 2d and particles on a circle
Physical YM is explicitly equivalent to a quantum mechanical model: free fermions on a circle
Can be checked by a partition function on a two-torus
GrossDouglas
YM in 2d and particles on a circleYM in 2d and particles on a circle
Physical YM is explicitly equivalent to a quantum mechanical model: free fermions on a circle
States are labelled by the partitions, for G=U(N)
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YM in 2d and particles on a circleYM in 2d and particles on a circle
For N=2 YM these free fermions on a circle
Label the vacua of the theory deformed by twisted superpotential W
YM in 2d and particles on a circleYM in 2d and particles on a circle
The fermions can be made interacting by adding a localized matter: for example a time-like Wilson loopin some representation V of the gauge group:
YM in 2d and particles on a circleYM in 2d and particles on a circle
One gets Calogero-Sutherland (spin) particles on a circle(1993-94) A.Gorsky,NN; J.Minahan,A.Polychronakos;
HistoryHistory
In 1997 G.Moore, NN and S.Shatashvili studied integrals over
various hyperkahler quotients, with the aim to understand
instanton integrals in four dimensional gauge theories
In 1997 G.Moore, NN and S.Shatashvili studied integrals over
various hyperkahler quotients, with the aim to understand
instanton integrals in four dimensional gauge theories
HistoryHistory
In 1997 G.Moore, NN and S.Shatashvili studied integrals over
various hyperkahler quotients, with the aim to understand
instanton integrals in four dimensional gauge theories
This eventually led to the derivation in 2002 of the Seiberg-Witten solution of N=2 d=4 theory
In 1997 G.Moore, NN and S.Shatashvili studied integrals over
various hyperkahler quotients, with the aim to understand
instanton integrals in four dimensional gauge theories
This eventually led to the derivation in 2002 of the Seiberg-Witten solution of N=2 d=4 theory
Inspired by the work of H.Nakajima
Yang-Mills-Higgs theoryYang-Mills-Higgs theory
Among various examples, MNS studied Hitchin’s moduli space MH
Among various examples, MNS studied Hitchin’s moduli space MH
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Yang-Mills-Higgs theoryYang-Mills-Higgs theory
Unlike the case of two-dimensionalYang-Mills theory where the moduli
space MG is compact,
Hitchin’s moduli space is non-compact
(it is roughly T*MG modulo subtleties) and the volume is infinite.
Unlike the case of two-dimensionalYang-Mills theory where the moduli
space MG is compact,
Hitchin’s moduli space is non-compact
(it is roughly T*MG modulo subtleties) and the volume is infinite.
Yang-Mills-Higgs theoryYang-Mills-Higgs theory
In order to cure this infnity in a reasonable way MNS used the U(1) symmetry of MH
In order to cure this infnity in a reasonable way MNS used the U(1) symmetry of MH
The volume becomes a DH-type expression:
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Where H is the Hamiltonian
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Yang-Mills-Higgs theoryYang-Mills-Higgs theoryUsing the supersymmetry and localization
the regularized volume of MH
was computed with the result
Using the supersymmetry and localization the regularized volume of MH
was computed with the result
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Yang-Mills-Higgs theoryYang-Mills-Higgs theory
Where the eigenvalues solve the equations: Where the eigenvalues solve the equations:
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YMH and NLSYMH and NLSThe experts would immediately recognise the
Bethe ansatz (BA) equations for the non-linear Schroedinger theory (NLS)
The experts would immediately recognise theBethe ansatz (BA) equations for
the non-linear Schroedinger theory (NLS)
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NLS = large spin limit of the SU(2) XXX spin chain
YMH and NLSYMH and NLS
Moreover the NLS Hamiltoniansare the 0-observables of the theory, like
Moreover the NLS Hamiltoniansare the 0-observables of the theory, like
The VEV of the observable =
The eigenvalue of the Hamiltonian
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YMH and NLSYMH and NLS
Since 1997 nothing came out of this result.
It could have been simply a coincidence.
…….
Since 1997 nothing came out of this result.
It could have been simply a coincidence.
…….
In 2006 A.Gerasimov and
S.Shatashvili have revived the subject
In 2006 A.Gerasimov and
S.Shatashvili have revived the subject
HistoryHistory
YMH and interacting particles
YMH and interacting particles
GS noticed that YMH theory viewed as TFT is equivalent to
the quantum Yang system: N particles on a circle with
delta-interaction:
GS noticed that YMH theory viewed as TFT is equivalent to
the quantum Yang system: N particles on a circle with
delta-interaction:
YMH and interacting particles
YMH and interacting particles
Thus:Thus: YM with the matter -- fermions with pair-wise
interaction
Thus:Thus: YM with the matter -- fermions with pair-wise
interaction
HistoryHistory
More importantly, GS suggested that TFT/QIS equivalence is much more
universal
More importantly, GS suggested that TFT/QIS equivalence is much more
universal
TodayToday
We shall rederive the result of MNS from a modern perspective
Generalize to cover virtually all BA soluble systems both with finite and infinite spin
Suggest natural extensions of the BA equations
We shall rederive the result of MNS from a modern perspective
Generalize to cover virtually all BA soluble systems both with finite and infinite spin
Suggest natural extensions of the BA equations
Hitchin equationsHitchin equations
Solutions can be viewed as the susy field configurations for
the N=2 gauged linear sigma model
Solutions can be viewed as the susy field configurations for
the N=2 gauged linear sigma model
For adjoint-valued linear fields
Hitchin equationsHitchin equations
The moduli space MH of solutions is a hyperkahler manifold
The integrals over MH are computed by the correlation functions of
an N=2 d=2 susy gauge theory
The moduli space MH of solutions is a hyperkahler manifold
The integrals over MH are computed by the correlation functions of
an N=2 d=2 susy gauge theory
Hitchin equationsHitchin equations
The kahler form on MH comes fromtwisted tree level superpotential
The epsilon-term comes from a twisted mass of the matter multiplet
The kahler form on MH comes fromtwisted tree level superpotential
The epsilon-term comes from a twisted mass of the matter multiplet
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GeneralizationGeneralization
Take an N=2 d=2 gauge theory with matter,
In some representation R of the gauge group G
Take an N=2 d=2 gauge theory with matter,
In some representation R of the gauge group G
GeneralizationGeneralization
Integrate out the matter fields, compute the effective (twisted)
super-potentialon the Coulomb branch
Integrate out the matter fields, compute the effective (twisted)
super-potentialon the Coulomb branch
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Mathematically speakingMathematically speaking
Consider the moduli space MR of R-Higgs pairswith gauge group G
Consider the moduli space MR of R-Higgs pairswith gauge group G
Up to the action of the complexified gauge group GC
Mathematically speakingMathematically speaking
Stability conditions:Stability conditions:
Up to the action of the compact gauge group G
Mathematically speakingMathematically speaking
Pushforward the unit class down to the moduli space MG of GC-bundles
Equivariantly with respect to the actionof the global symmetry group K on MR
Pushforward the unit class down to the moduli space MG of GC-bundles
Equivariantly with respect to the actionof the global symmetry group K on MR
Mathematically speakingMathematically speaking
The pushforward can be expressed in terms of the Donaldson-like classes of
the moduli space MG
2-observables and 0-observables
The pushforward can be expressed in terms of the Donaldson-like classes of
the moduli space MG
2-observables and 0-observables
Mathematically speakingMathematically speaking
The pushforward can be expressed in terms of the Donaldson-like classes of
the moduli space MG
2-observables and 0-observables
The pushforward can be expressed in terms of the Donaldson-like classes of
the moduli space MG
2-observables and 0-observables
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Mathematically speakingMathematically speaking
The masses are the equivariant parameters
For the global symmetry group K
The masses are the equivariant parameters
For the global symmetry group K
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Vacua of the gauge theoryVacua of the gauge theory
Due to quantization of the gauge fluxDue to quantization of the gauge flux
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For G = U(N)
Vacua of the gauge theoryVacua of the gauge theory
Equations familiar from yesterday’s lectureEquations familiar from yesterday’s lecture
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For G = U(N)
partitions
Vacua of the gauge theoryVacua of the gauge theory
Familiar example: CPN modelFamiliar example: CPN model(N+1) chiral multiplet of charge +1
Qi i=1, … , N+1U(1) gauge group
N+1 vacuum
Field content:
Effective superpotential:
Vacua of gauge theoryVacua of gauge theory
Gauge group: G=U(N)Matter chiral multiplets: 1 adjoint, mass fundamentals, massanti-fundamentals, mass
Gauge group: G=U(N)Matter chiral multiplets: 1 adjoint, mass fundamentals, massanti-fundamentals, mass Field content:
Another example:
Vacua of gauge theoryVacua of gauge theory
Effective superpotential:
Vacua of gauge theoryVacua of gauge theory
Equations for vacua:
Vacua of gauge theoryVacua of gauge theory
Non-anomalous case:Redefine:
Vacua of gauge theoryVacua of gauge theory
Vacua:
Gauge theory -- spin chainGauge theory -- spin chain
Identical to the Bethe ansatz equations for spin XXX magnet:
Gauge theory -- spin chainGauge theory -- spin chain
Vacua = eigenstates of the Hamiltonian:
Table of dualitiesTable of dualities
XXX spin chain SU(2)L spinsN excitations
XXX spin chain SU(2)L spinsN excitations
U(N) d=2 N=2 Chiral multiplets:1 adjointL fundamentalsL anti-fund.
U(N) d=2 N=2 Chiral multiplets:1 adjointL fundamentalsL anti-fund.
Special masses!
Table of dualities: mathematically speaking
Table of dualities: mathematically speaking
XXX spin chain SU(2)L spinsN excitations
XXX spin chain SU(2)L spinsN excitations
(Equivariant)Intersection theory on MR for
(Equivariant)Intersection theory on MR for
Table of dualitiesTable of dualities
XXZ spin chain SU(2)L spinsN excitations
XXZ spin chain SU(2)L spinsN excitations
U(N) d=3 N=1Compactified on a circle Chiral multiplets:1 adjointL fundamentalsL anti-fund.
U(N) d=3 N=1Compactified on a circle Chiral multiplets:1 adjointL fundamentalsL anti-fund.
Table of dualities: mathematically speaking
Table of dualities: mathematically speaking
XXZ spin chain SU(2)L spinsN excitations
XXZ spin chain SU(2)L spinsN excitations
Equivariant K-theory of the
moduli space MR
Equivariant K-theory of the
moduli space MR
Table of dualitiesTable of dualities
XYZ spin chain SU(2), L = 2N
spinsN excitations
XYZ spin chain SU(2), L = 2N
spinsN excitations
U(N) d=4 N=1Compactified on a 2-torus
= elliptic curve E Chiral multiplets:1 adjointL = 2N fundamentalsL = 2N anti-fund.
U(N) d=4 N=1Compactified on a 2-torus
= elliptic curve E Chiral multiplets:1 adjointL = 2N fundamentalsL = 2N anti-fund.
Masses = wilson loops of the flavour group= points on the Jacobian of E
Table of dualities: mathematically speaking
Table of dualities: mathematically speaking
XYZ spin chain SU(2), L = 2N
spinsN excitations
XYZ spin chain SU(2), L = 2N
spinsN excitations
Elliptic genus of the moduli space MR
Elliptic genus of the moduli space MR
Masses = K bundle over E= points on the BunK of E
Table of dualitiesTable of dualities
It is remarkable that the spin chain hasprecisely those generalizations:
rational (XXX), trigonometric (XXZ) and elliptic (XYZ)
that can be matched to the 2, 3, and 4 dim cases.
It is remarkable that the spin chain hasprecisely those generalizations:
rational (XXX), trigonometric (XXZ) and elliptic (XYZ)
that can be matched to the 2, 3, and 4 dim cases.
Algebraic Bethe AnsatzAlgebraic Bethe Ansatz
The spin chain is solved algebraically using certain operators,
Which obey exchange commutation relations
The spin chain is solved algebraically using certain operators,
Which obey exchange commutation relations
Faddeev et al.
Faddeev-Zamolodchikov algebra…
Algebraic Bethe AnsatzAlgebraic Bethe Ansatz
The eigenvectors, Bethe vectors, are obtained by applying these
operators to the « fake » vacuum.
The eigenvectors, Bethe vectors, are obtained by applying these
operators to the « fake » vacuum.
ABA vs GAUGE THEORYABA vs GAUGE THEORYFor the spin chain it is natural to fix L = total
number of spinsand consider various N = excitation levels
In the gauge theory context N is fixed.
For the spin chain it is natural to fix L = total number of spins
and consider various N = excitation levels
In the gauge theory context N is fixed.
ABA vs GAUGE THEORYABA vs GAUGE THEORYHowever, if the theory is embedded
into string theory via brane realization
then changing N is easy: bring in an extra brane.
However, if the theory is embedded into string theory via brane
realization then changing N is easy: bring in an extra brane.
Hanany-Hori’02
ABA vs GAUGE THEORYABA vs GAUGE THEORY
Mathematically speaking We claim that the Algebraic Bethe Ansatz
is most naturally related to the derived category of the category of coherent
sheaves on some local CY
Mathematically speaking We claim that the Algebraic Bethe Ansatz
is most naturally related to the derived category of the category of coherent
sheaves on some local CY
ABA vs STRING THEORYABA vs STRING THEORY
THUS:
B is for BRANE!
THUS:
B is for BRANE!
is for location!
More general spin chainsMore general spin chains
The SU(2) spin chain has generalizations to
other groups and representations.
I quote the corresponding Bethe ansatz equations
from N.Reshetikhin
The SU(2) spin chain has generalizations to
other groups and representations.
I quote the corresponding Bethe ansatz equations
from N.Reshetikhin
General groups/repsGeneral groups/reps
For simply-laced group H of rank rFor simply-laced group H of rank r
General groups/repsGeneral groups/reps
For simply-laced group H of rank rFor simply-laced group H of rank r
Label representations of the Yangian of H A.N.Kirillov-N.Reshetikhin modules
Cartan matrix of H
General groups/repsfrom GAUGE THEORYGeneral groups/repsfrom GAUGE THEORY
Take the Dynkin diagram corresponding to H A simply-laced group of rank r
Take the Dynkin diagram corresponding to H A simply-laced group of rank r
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
SymmetriesSymmetries
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
SymmetriesSymmetries
QUIVER GAUGE THEORYCharged matter
QUIVER GAUGE THEORYCharged matter
Adjoint chiral multiplet
Fundamental chiral multiplet
Anti-fundamental chiral multiplet
Bi-fundamental chiral multiplet
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
Matter fields: adjoints Matter fields: adjoints
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
Matter fields: fundamentals+anti-fundamentals
Matter fields: fundamentals+anti-fundamentals
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
Matter fields: bi-fundamentals Matter fields: bi-fundamentals
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
Quiver gauge theory: full content Quiver gauge theory: full content
QUIVER GAUGE THEORY: MASSES
QUIVER GAUGE THEORY: MASSES
Adjoints Adjoints
i
QUIVER GAUGE THEORY: MASSES
QUIVER GAUGE THEORY: MASSES
FundamentalsAnti-fundamentals FundamentalsAnti-fundamentals
i
a = 1, …. , Li
QUIVER GAUGE THEORY: MASSES
QUIVER GAUGE THEORY: MASSES
Bi-fundamentals Bi-fundamentals
i j
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
What is so special about these masses? What is so special about these masses?
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
From the gauge theory point of view nothing special…..
From the gauge theory point of view nothing special…..
QUIVER GAUGE THEORYQUIVER GAUGE THEORY
The mass puzzle!The mass puzzle!
The mass puzzleThe mass puzzle
The Bethe ansatz -- like equationsThe Bethe ansatz -- like equations
Can be written for an arbitrary matrix
The mass puzzleThe mass puzzle
However the Yangian symmetry Y(H) would get replaced by some ugly infinite-
dimensional « free » algreba without nice representations
However the Yangian symmetry Y(H) would get replaced by some ugly infinite-
dimensional « free » algreba without nice representations
The mass puzzleThe mass puzzle
Therefore we conclude that our choice of masses is dictated by the hidden symmetry -- that of
the dual spin chain
Therefore we conclude that our choice of masses is dictated by the hidden symmetry -- that of
the dual spin chain
The Standard Model has many free parameters
The Standard Model has many free parameters
Among them are the fermion masses Is there a (hidden) symmetry
principle behind them?
Among them are the fermion masses Is there a (hidden) symmetry
principle behind them?
The Standard Model has many free parameters
The Standard Model has many free parameters
In the supersymmetric modelswe considered
the mass tuning can be « explained »
using a duality to some quantum integrable system
In the supersymmetric modelswe considered
the mass tuning can be « explained »
using a duality to some quantum integrable system
Further generalizations:Superpotential
from prepotential
Further generalizations:Superpotential
from prepotential
Tree level part
Induced by twist
Flux superpotential(Losev,NN, Shatashvili’97)
The N=2* theory on R2 X S2
Superpotential from prepotentialSuperpotential
from prepotential
Magnetic flux
Electric flux
In the limit of vanishing S2 the magnetic flux should vanish
Instanton corrected BA equations
Instanton corrected BA equations
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Effective S-matrix contains 2-body, 3-body, … interactions
Instanton corrected BA equations
Instanton corrected BA equations
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Instanton corrected BA equations
Instanton corrected BA equations
The prepotential of the low-energy effective theoryIs governed by a classical (holomorphic) integrable system
Donagi-Witten’95
Liouville tori = Jacobians of Seiberg-Witten curves
Classical integrable system
vsQuantum integrable
system
Classical integrable system
vsQuantum integrable
systemThat system is quantized when the gauge theory is subject to
the Omega-background
NN’02NN,Okounkov’03Braverman’03
Our quantum system is different!
Blowing up the two-sphereBlowing up the two-sphere
Wall-crossing phenomena(new states, new solutions)Wall-crossing phenomena
(new states, new solutions)
Something for the future
Naturalness of our quiversNaturalness of our quivers
Somewhat unusual matter contentBranes at orbifolds typically lead to
smth like
Somewhat unusual matter contentBranes at orbifolds typically lead to
smth like
Naturalness of our quiversNaturalness of our quivers
This picture would arise in the sa
(i) 0
limit
This picture would arise in the sa
(i) 0
limit BA for QCD
Faddeev-Korchemsky’94
Naturalness of our quiversNaturalness of our quivers
Other quivers? Other quivers?
Naturalness of our quiversNaturalness of our quivers
Possibly with the help of K.Saito’s construction
Possibly with the help of K.Saito’s construction
CONCLUSIONSCONCLUSIONS
1. We found the Bethe Ansatz equations are the equations describing the vacuum configurations of certain quiver gauge theories in two dimensions
2. The duality to the spin chain requires certain relations between the masses of the matter fields to be obeyed. This could have phenomenological consequences.
1. We found the Bethe Ansatz equations are the equations describing the vacuum configurations of certain quiver gauge theories in two dimensions
2. The duality to the spin chain requires certain relations between the masses of the matter fields to be obeyed. This could have phenomenological consequences.
CONCLUSIONSCONCLUSIONS
3. The algebraic Bethe ansatz seems to provide a realization of the brane creation operators -- something of major importance both for topological and physical string theories
4. Obviously this is a beginning of a beautiful story….
3. The algebraic Bethe ansatz seems to provide a realization of the brane creation operators -- something of major importance both for topological and physical string theories
4. Obviously this is a beginning of a beautiful story….