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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; 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 theories with four supersymmetries N=1 d=4 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. 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 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 MG of flat G-connections on a Riemann surface Yang-Mills theories in 2d N=2 super-Yang-Mills theory QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Field content: Yang-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 Yang-Mills theories in 2d Physical YM QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 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: QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Yang-Mills theories in 2d Two dimensional Yang-Mills partition function is given by the explicit sum QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Yang-Mills theories in 2d In the limit Quic kTime™ and a TIFF (Unc ompres sed) decompress or are needed to see this picture. the partition function computes the volume of MG QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Yang-Mills theories in 2d Witten’s approach: add twisted superpotential and its conjugate QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Yang-Mills theories in 2d Take a limit QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. In the limit the fields QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. are infinitely massive and can be integrated out: one is left with the field content of the physical YM theory QuickTi me™ and a TIFF ( Uncompressed) decompr essor are needed to see thi s picture. Yang-Mills theories in 2d 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) 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) QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 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 (1993-94) 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 History In 1997 G.Moore, NN and S.Shatashvili studied integrals over Inspired by the work of H.Nakajima 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 Yang-Mills-Higgs theory Among various examples, MNS studied Hitchin’s moduli space MH QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Yang-Mills-Higgs theory Unlike the case of two-dimensional Yang-Mills theory where the moduli space MG is compact, Hitchin’s moduli space is noncompact (it is roughly T*MG 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 MH QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. The volume becomes a DH-type expression: QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Where H is the Hamiltonian QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Yang-Mills-Higgs theory Using the supersymmetry and localization the regularized volume of MH was computed with the result QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Yang-Mills-Higgs theory Where the eigenvalues solve the equations: QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. YMH and NLS The experts would immediately recognise the Bethe ansatz (BA) equations for the non-linear Schroedinger theory (NLS) QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 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 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 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. ……. History In 2006 A.Gerasimov and S.Shatashvili have revived the subject 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: YMH and interacting particles Thus: YM with the matter -fermions with pair-wise interaction History 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 Hitchin equations 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 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 equations The kahler form on MH comes from twisted tree level superpotential The epsilon-term comes from a twisted mass of the matter multiplet QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Generalization 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 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Mathematically speaking Consider the moduli space MR of R-Higgs pairs with gauge group G Up to the action of the complexified gauge group GC 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 MG of GC-bundles Equivariantly with respect to the action of the global symmetry group K on MR Mathematically speaking The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space MG 2-observables and 0-observables Mathematically speaking The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space MG 2-observables and 0-observables QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Mathematically speaking The masses are the equivariant parameters For the global symmetry group K QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Vacua of the gauge theory For G = U(N) QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Due to quantization of the gauge flux QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Vacua of the gauge theory For G = U(N) QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Equations familiar from yesterday’s lecture QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. partitions Vacua of the gauge theory Familiar example: CPN model Field content: (N+1) chiral multiplet of charge +1 Qi i=1, … , N+1 U(1) gauge group Effective superpotential: N+1 vacuum Vacua of gauge theory Another example: Field Gauge group: content: G=U(N) Matter chiral multiplets: 1 adjoint, mass fundamentals, mass anti-fundamentals, mass Vacua of gauge theory Effective superpotential: Vacua of gauge theory Equations for vacua: Vacua of gauge theory Nonanomalo us case: Redefine: Vacua of gauge theory Vacua: Gauge theory -- spin chain Identical to the Bethe ansatz equation s for Gauge theory -- spin chain Vacua = eigensta tes of the Hamilton ian: Table of dualities XXX spin chain SU(2) L spins N excitations 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 (Equivariant) Intersection theory on MR for Table of dualities 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. Table of dualities: mathematically speaking XXZ spin chain SU(2) L spins N excitations Equivariant K-theory of the moduli space MR Table of dualities 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. 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 Elliptic genus of the moduli space MR Masses = K bundle over E = points on the BunK 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. Algebraic Bethe Ansatz Faddeev et al. The spin chain is solved algebraically using certain operators, Which obey exchange commutation relations 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. 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. 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 ABA vs STRING THEORY 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 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 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 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 i a = 1, …. , Li 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? 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 Further generalizations: Superpotential from prepotential Tree level part Induced by twist The N=2* theory on R2 X S2 Flux superpotential (Losev,NN, Shatashvili’97) Superpotential from prepotential Magnetic flux Electric flux In the limit of vanishing S2 the magnetic flux should vanish Instanton corrected BA equations QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Effective S-matrix contains 2-body, 3-body, … interactions Instanton corrected BA equations QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 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) Something for the future Naturalness of our quivers Somewhat unusual matter content Branes at orbifolds typically lead to smth like Naturalness of our quivers This picture would arise in the sa(i) 0 BA for QCD Faddeev-Korchemsky’94 limit Naturalness of our quivers Other quivers? Naturalness of our quivers Possibly with the help of K.Saito’s construction CONCLUSIONS 1. 2. We found the Bethe Ansatz equations are the equations describing the vacuum configurations of certain quiver gauge theories in two dimensions 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….