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 on NN, S.Shatashvili, to appear Prior work: E.Witten, 1992; A.Gorsky, NN; J.Minahan, A.Polychronakos; M.Douglas; ~ ; A.Gerasimov ~1993; G.Moore, NN, S.Shatashvili ~ ; A.Losev, NN, S.Shatashvili ~ ; A.Gerasimov, S.Shatashvili ~ NN, S.Shatashvili, to appear Prior work: E.Witten, 1992; A.Gorsky, NN; J.Minahan, A.Polychronakos; M.Douglas; ~ ; A.Gerasimov ~1993; G.Moore, NN, S.Shatashvili ~ ; A.Losev, NN, S.Shatashvili ~ ; A.Gerasimov, S.Shatashvili ~

We are going to relate 2,3, and 4 dimensional susy gauge theories with four supersymmetries N=1 d=4 And quantum integrable systems soluble by Bethe Ansatz techniques. And quantum integrable systems soluble 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 And quantum integrable systems soluble by Bethe Ansatz techniques. And quantum integrable systems soluble 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 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 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 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 M G 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 M G of flat G-connections on a Riemann surface

Yang-Mills theories in 2d N=2 super-Yang-Mills theory Field content:

Yang-Mills theories in 2d (2) Physical YM = N=0 Yang-Mills theory, with gauge group G; The moduli space M G 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 M G 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 2d Physical YM Field content:

Yang-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:

Yang-Mills theories in 2d Two dimensional Yang-Mills partition function is given by the explicit sum

Yang-Mills theories in 2d In the limit the partition function computes the volume of M G In the limit the partition function computes the volume of M G

Yang-Mills theories in 2d Witten’s approach: add twisted superpotential and its conjugate

Yang-Mills theories in 2d 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

Yang-Mills theories in 2d Both physical and cohomological Yang-Mills theories define topological field theories (TFT) Both physical and cohomological Yang-Mills theories define topological field theories (TFT)

Yang-Mills theories in 2d Both 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 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 Gross Douglas

YM 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)

YM 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 circle The fermions can be made interacting by adding a localized matter: for example a time-like Wilson loop in some representation V of the gauge group:

YM in 2d and particles on a circle One gets Calogero-Sutherland (spin) particles on a circle ( ) A.Gorsky,NN; J.Minahan,A.Polychronakos;

History 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

History 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 theory Among various examples, MNS studied Hitchin’s moduli space M H

Yang-Mills-Higgs theory Unlike the case of two-dimensional Yang-Mills theory where the moduli space M G is compact, Hitchin’s moduli space is non- compact (it is roughly T * M G modulo subtleties) and the volume is infinite. Unlike the case of two-dimensional Yang-Mills theory where the moduli space M G is compact, Hitchin’s moduli space is non- compact (it is roughly T * M G modulo subtleties) and the volume is infinite.

Yang-Mills-Higgs theory In order to cure this infnity in a reasonable way MNS used the U(1) symmetry of M H The volume becomes a DH-type expression: Where H is the Hamiltonian

Yang-Mills-Higgs theory Using the supersymmetry and localization the regularized volume of M H was computed with the result Using the supersymmetry and localization the regularized volume of M H was computed with the result

Yang-Mills-Higgs theory Where the eigenvalues solve the equations:

YMH and NLS The experts would immediately recognise the Bethe ansatz (BA) equations for the non-linear Schroedinger theory (NLS) The experts would immediately recognise the Bethe ansatz (BA) equations for the non-linear Schroedinger theory (NLS) NLS = large spin limit of the SU(2) XXX spin chain

YMH and NLS Moreover the NLS Hamiltonians are the 0-observables of the theory, like Moreover the NLS Hamiltonians are the 0-observables of the theory, like The VEV of the observable = The eigenvalue of the Hamiltonian

YMH 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 History

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 Thus: Thus: YM with the matter -- fermions with pair-wise interaction

History More importantly, GS suggested that TFT/QIS equivalence is much more universal More importantly, GS suggested that TFT/QIS equivalence is much more universal

Today 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 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 equations The moduli space M H of solutions is a hyperkahler manifold The integrals over M H are computed by the correlation functions of an N=2 d=2 susy gauge theory The moduli space M H of solutions is a hyperkahler manifold The integrals over M H are computed by the correlation functions of an N=2 d=2 susy gauge theory

Hitchin equations The kahler form on M H comes from twisted tree level superpotential The epsilon-term comes from a twisted mass of the matter multiplet The kahler form on M H comes from twisted tree level superpotential The epsilon-term comes from a twisted mass of the matter multiplet

Generalization 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

Generalization Integrate out the matter fields, compute the effective (twisted) super-potential on the Coulomb branch Integrate out the matter fields, compute the effective (twisted) super-potential on the Coulomb branch

Mathematically speaking Consider the moduli space M R of R -Higgs pairs with gauge group G Consider the moduli space M R of R -Higgs pairs with gauge group G Up to the action of the complexified gauge group G C

Mathematically speaking Stability conditions: Up to the action of the compact gauge group G

Mathematically speaking Pushforward the unit class down to the moduli space M G of G C -bundles Equivariantly with respect to the action of the global symmetry group K on M R Pushforward the unit class down to the moduli space M G of G C -bundles Equivariantly with respect to the action of the global symmetry group K on M R

Mathematically speaking The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space M G 2-observables and 0-observables The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space M G 2-observables and 0-observables

Mathematically speaking The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space M G 2-observables and 0-observables The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space M G 2-observables and 0-observables

Mathematically 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

Vacua of the gauge theory Due to quantization of the gauge flux For G = U(N)

Vacua of the gauge theory Equations familiar from yesterday’s lecture For G = U(N) partitions

Vacua of the gauge theory Familiar example: CP N model (N+1) chiral multiplet of charge +1 Q i i=1, …, N+1 U(1) gauge group N+1 vacuum Field content: Effective superpotential:

Vacua of gauge theory Gauge group: G=U(N) Matter chiral multiplets: 1 adjoint, mass fundamentals, mass anti-fundamentals, mass Gauge group: G=U(N) Matter chiral multiplets: 1 adjoint, mass fundamentals, mass anti-fundamentals, mass Field content: Another example:

Vacua of gauge theory Effective superpotential:

Vacua of gauge theory Equations for vacua:

Vacua of gauge theory Non- anomalo us case: Redefine:

Vacua of gauge theory Vacua:

Gauge theory -- spin chain Identical to the Bethe ansatz equation s for spin XXX magnet:

Gauge theory -- spin chain Vacua = eigensta tes of the Hamilton ian:

Table of dualities XXX spin chain SU(2) L spins N excitations XXX spin chain SU(2) L spins N excitations U(N) d=2 N=2 Chiral multiplets: 1 adjoint L fundamentals L anti-fund. U(N) d=2 N=2 Chiral multiplets: 1 adjoint L fundamentals L anti-fund. Special masses!

Table of dualities: mathematically speaking XXX spin chain SU(2) L spins N excitations XXX spin chain SU(2) L spins N excitations (Equivariant) Intersection theory on M R for (Equivariant) Intersection theory on M R for

Table of dualities XXZ spin chain SU(2) L spins N excitations XXZ spin chain SU(2) L spins N excitations U(N) d=3 N=1 Compactified on a circle Chiral multiplets: 1 adjoint L fundamentals L anti-fund. U(N) d=3 N=1 Compactified on a circle Chiral multiplets: 1 adjoint L fundamentals L anti-fund.

Table of dualities: mathematically speaking XXZ spin chain SU(2) L spins N excitations XXZ spin chain SU(2) L spins N excitations Equivariant K-theory of the moduli space M R

Table of dualities XYZ spin chain SU(2), L = 2N spins N excitations XYZ spin chain SU(2), L = 2N spins N excitations U(N) d=4 N=1 Compactified on a 2-torus = elliptic curve E Chiral multiplets: 1 adjoint L = 2N fundamentals L = 2N anti-fund. U(N) d=4 N=1 Compactified on a 2-torus = elliptic curve E Chiral multiplets: 1 adjoint L = 2N fundamentals L = 2N anti-fund. Masses = wilson loops of the flavour group = points on the Jacobian of E

Table of dualities: mathematically speaking XYZ spin chain SU(2), L = 2N spins N excitations XYZ spin chain SU(2), L = 2N spins N excitations Elliptic genus of the moduli space M R Masses = K bundle over E = points on the Bun K of E

Table of dualities It is remarkable that the spin chain has precisely 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 has precisely those generalizations: rational (XXX), trigonometric (XXZ) and elliptic (XYZ) that can be matched to the 2, 3, and 4 dim cases.

Algebraic 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 Ansatz The eigenvectors, Bethe vectors, are obtained by applying these operators to the « fake » vacuum.

ABA vs GAUGE THEORY 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. 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 THEORY However, 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 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 THEORY THUS: B is for BRANE! THUS: B is for BRANE! is for location!

More 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/reps For simply-laced group H of rank r

General groups/reps For 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/reps from 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 THEORY Symmetries

QUIVER GAUGE THEORY Symmetries

QUIVER GAUGE THEORY Charged matter Adjoint chiral multiplet Fundamental chiral multiplet Anti-fundamental chiral multiplet Bi-fundamental chiral multiplet

QUIVER GAUGE THEORY Matter fields: adjoints

QUIVER GAUGE THEORY Matter fields: fundamentals+anti-fundamentals Matter fields: fundamentals+anti-fundamentals

QUIVER GAUGE THEORY Matter fields: bi-fundamentals

QUIVER GAUGE THEORY Quiver gauge theory: full content

QUIVER GAUGE THEORY: MASSES Adjoints i

QUIVER GAUGE THEORY: MASSES Fundamentals Anti-fundamentals Fundamentals Anti-fundamentals i a = 1, …., L i

QUIVER GAUGE THEORY: MASSES Bi-fundamentals i j

QUIVER GAUGE THEORY What is so special about these masses?

QUIVER GAUGE THEORY From the gauge theory point of view nothing special…..

QUIVER GAUGE THEORY The mass puzzle!

The mass puzzle The Bethe ansatz -- like equations Can be written for an arbitrary matrix

The mass puzzle However the Yangian symmetry Y(H) would get replaced by some ugly infinite-dimensional « free » algreba without nice representations

The mass puzzle 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?

The Standard Model has many free parameters In the supersymmetric models we considered the mass tuning can be « explained » using a duality to some quantum integrable system In the supersymmetric models we considered the mass tuning can be « explained » using a duality to some quantum integrable system

Further generalizations: Superpotential from prepotential Tree level part Induced by twist Flux superpotential (Losev,NN, Shatashvili’97) The N=2* theory on R 2 X S 2

Superpotential from prepotential Magnetic flux Electric flux In the limit of vanishing S 2 the magnetic flux should vanish

Instanton corrected BA equations Effective S-matrix contains 2-body, 3-body, … interactions

Instanton corrected BA equations

The prepotential of the low-energy effective theory Is governed by a classical (holomorphic) integrable system Donagi-Witten’95 Liouville tori = Jacobians of Seiberg-Witten curves

Classical integrable system vs Quantum integrable system That system is quantized when the gauge theory is subject to the Omega-background NN’02 NN,Okounkov’03 Braverman’03 Our quantum system is different!

Blowing 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 quivers Somewhat unusual matter content Branes at orbifolds typically lead to smth like Somewhat unusual matter content Branes at orbifolds typically lead to smth like

Naturalness of our quivers This picture would arise in the s a (i)  0 limit This picture would arise in the s a (i)  0 limit BA for QCD Faddeev-Korchemsky’94

Naturalness of our quivers Other quivers?

Naturalness of our quivers Possibly with the help of K.Saito’s construction

CONCLUSIONS 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.

CONCLUSIONS 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….