Two Dimensional Gauge Theories and Quantum Integrable Systems

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.


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



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

Field content: QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.


(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


Physical YM Physical YM

Field content:QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.


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:

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.


Two dimensional Yang-Mills partition function is given by the explicit sum

Two dimensional Yang-Mills partition function is given by the explicit sum




In the limit

the partition function computes the volume of MG

In the limit

the partition function computes the volume of MG






Witten’s approach: add twisted superpotential and its conjugate

Witten’s approach: add twisted superpotential and its conjugate




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





QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.


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


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)




For N=2 YM these free fermions on a circle

Label the vacua of the theory deformed by twisted superpotential W


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:


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




HistoryHistory




This eventually led to the derivation in 2002 of the Seiberg-Witten solution of N=2 d=4 theory




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




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.


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:



Where H is the Hamiltonian





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






Where the eigenvalues solve the equations: Where the eigenvalues solve the equations:



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)



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



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


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:



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


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


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



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



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


Stability conditions:Stability conditions:

Up to the action of the compact gauge group G


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


The pushforward can be expressed in terms of the Donaldson-like classes of

the moduli space MG

2-observables and 0-observables


the moduli space MG




the moduli space MG



the moduli space MG





The masses are the equivariant parameters

For the global symmetry group K

The masses are the equivariant parameters

For the global symmetry group K



Vacua of the gauge theoryVacua of the gauge theory

Due to quantization of the gauge fluxDue to quantization of the gauge flux




For G = U(N)


Equations familiar from yesterday’s lectureEquations familiar from yesterday’s lecture




For G = U(N)

partitions


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:


Effective superpotential:


Equations for vacua:


Non-anomalous case:Redefine:


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


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




(Equivariant)Intersection theory on MR for

(Equivariant)Intersection theory on MR for


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.





Equivariant K-theory of the

moduli space MR

Equivariant K-theory of the

moduli space MR


XYZ spin chain SU(2), L = 2N

spinsN excitations


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




spinsN excitations


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


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


SymmetriesSymmetries

QUIVER GAUGE THEORYCharged matter

QUIVER GAUGE THEORYCharged matter

Adjoint chiral multiplet

Fundamental chiral multiplet

Anti-fundamental chiral multiplet

Bi-fundamental chiral multiplet


Matter fields: adjoints Matter fields: adjoints


Matter fields: fundamentals+anti-fundamentals

Matter fields: fundamentals+anti-fundamentals


Matter fields: bi-fundamentals Matter fields: bi-fundamentals


Quiver gauge theory: full content Quiver gauge theory: full content

QUIVER GAUGE THEORY: MASSES


Adjoints Adjoints

i



FundamentalsAnti-fundamentals FundamentalsAnti-fundamentals

i

a = 1, …. , Li



Bi-fundamentals Bi-fundamentals

i j


What is so special about these masses? What is so special about these masses?


From the gauge theory point of view nothing special…..

From the gauge theory point of view nothing special…..


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


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


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


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?



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




Effective S-matrix contains 2-body, 3-body, … interactions







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


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


Other quivers? Other 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….

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Two Dimensional Gauge Theories and Quantum Integrable Systems