wo Dimensional Gauge Theoriesand Quantum Integrable Sys：两维规范理论量子可积系统.ppt

,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,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,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:,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 MG,Yang-Mills theories in 2d,Wittens 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,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),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 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 Hitchins moduli space MH,Yang-Mills-Higgs theory,Unlike the case of two-dimensional Yang-Mills theory where the moduli space MG is compact, Hitchins moduli space is non-compact (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,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 MH 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),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,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. .,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:,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,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,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,Mathematically speaking,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 yesterdays lecture,For G = U(N),partitions,Vacua of the gauge theory,Familiar example: CPN model,(N+1) chiral multiplet of charge +1 Qi 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,Field content:,Another example:,Vacua of gauge theory,Effective superpotential:,Vacua of gauge theory,Equations for vacua:,Vacua of gauge theory,Non-anomalous case:,Redefine:,Vacua of gauge theory,Vacua:,Gauge theory - spin chain,Identical to the Bethe ansatz equations for spin XXX magnet:,Gauge theory - spin chain,Vacua = eigenstates of the Hamiltonian:,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,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.,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-Hori02,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,Flux superpotential (Losev,NN, Shatashvili97),The N=2* theory on R2 X S2,Superpotential 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,Instanton corrected BA equations,Instanton corrected BA equ