1、Two Dimensional Gauge Theoriesand Quantum Integrable SystemsNikita Nekrasov IHESImperial College April 10,2008Based onNN,S.Shatashvili,to appearPrior 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.Shatashv
2、ili 1997-1998;A.Gerasimov,S.Shatashvili 2006-2007We are going to relate 2,3,and 4 dimensional susy gauge theorieswith four supersymmetries N=1 d=4And quantum integrable systemssoluble by Bethe Ansatz techniques.Mathematically speaking,the cohomology,K-theory and elliptic cohomology of various gauge
3、theory moduli spaces,like moduli of flat connections and instantonsAnd 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(pre-)HistoryIn 1992 E.Witten studied two dimensional Yang-Mills theory with
4、 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 theoriesYang-Mills theories in 2d(1)Cohomological YM=twisted N=2 super-Yang-Mills theory,with gauge group G,whose BPS(or TFT)sector is
5、 related to the intersection theory on the moduli space MG of flat G-connections on a Riemann surfaceYang-Mills theories in 2dN=2 super-Yang-Mills theoryField content:Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Yang-Mills theories in 2d(2)Physical YM=N=0 Yan
6、g-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 2dPhysical YM Field content:Q ui ckTi m e?and aTIFF(U ncom pressed)decom pressorare needed to
7、 see thi s pi cture.Yang-Mills theories in 2dWitten found a way to map the BPS sector of the N=2 theory to the N=0 theory.The result is:Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Yang-Mills theories in 2dTwo dimensional Yang-Mills partition function is give
8、n by the explicit sumQ ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Yang-Mills theories in 2dIn the limit the partition function computes the volume of MG Q ui ckTi m e?and aTI FF(U ncom p ressed)decom pressorare needed to see thi s pi cture.Q ui ckTi m e?and a
9、TI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Yang-Mills theories in 2dWittens approach:add twisted superpotential and its conjugate Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Yang-Mills theories in 2dTake a limit In the limit the field
10、s are infinitely massive and can be integrated out:one is left with the field content of the physical YM theory Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Q ui ckTi m e?an d aTIFF(U ncom p ressed)decom pressorare needed to see thi s pi cture.Q ui ckTi m e?a
11、nd aTIFF(U nco m pressed)decom pressorare needed to see thi s pi cture.Yang-Mills theories in 2dBoth physical and cohomological Yang-Millstheories define topological field theories(TFT)Yang-Mills theories in 2dBoth physical and cohomological Yang-Millstheories define topological field theories(TFT)V
12、acuum states+deformations=quantum mechanicsYM in 2d and particles on a circlePhysical YM is explicitly equivalent to a quantum mechanical model:free fermions on a circleCan be checked by a partition function on a two-torusGrossDouglasYM in 2d and particles on a circlePhysical YM is explicitly equiva
13、lent to a quantum mechanical model:free fermions on a circleStates are labelled by the partitions,for G=U(N)Q ui ckTi m e?and aTIFF(U ncom pressed)decom pressorare needed to see thi s pi cture.YM in 2d and particles on a circleFor N=2 YM these free fermions on a circleLabel the vacua of the theory d
14、eformed by twisted superpotential WYM in 2d and particles on a circleThe 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 circleOne gets Calogero-Sutherland(spin)particles on a cir
15、cle(1993-94)A.Gorsky,NN;J.Minahan,A.Polychronakos;HistoryIn 1997 G.Moore,NN and S.Shatashvili studied integrals over various hyperkahler quotients,with the aim to understand instanton integrals in four dimensional gauge theoriesHistoryIn 1997 G.Moore,NN and S.Shatashvili studied integrals over vario
16、us hyperkahler quotients,with the aim to understand instanton integrals in four dimensional gauge theoriesThis eventually led to the derivation in 2002 of the Seiberg-Witten solution of N=2 d=4 theoryInspired by the work of H.NakajimaYang-Mills-Higgs theoryAmong various examples,MNS studied Hitchins
17、 moduli space MHQ ui ckTi m e?and aTI FF(U ncom pressed)deco m pressorare needed to see thi s p i cture.Yang-Mills-Higgs theoryUnlike the case of two-dimensionalYang-Mills theory where the moduli space MG is compact,Hitchins moduli space is non-compact(it is roughly T*MG modulo subtleties)and the vo
18、lume is infinite.Yang-Mills-Higgs theoryIn order to cure this infnity in a reasonable way MNS used the U(1)symmetry of MHThe volume becomes a DH-type expression:Q ui ckTi m e?and aTIFF(U ncom pressed)decom presso rare needed to see thi s pi cture.Where H is the HamiltonianQ ui ckTi m e?and aTI FF(U
19、ncom pressed)decom pressorare needed to see thi s pi cture.Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s picture.Yang-Mills-Higgs theoryUsing the supersymmetry and localization the regularized volume of MH was computed with the resultQ ui ckTi m e?and aTI FF(U ncom pre
20、ssed)deco m pressorare needed to see thi s p i cture.Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Yang-Mills-Higgs theory Where the eigenvalues solve the equations:Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.YMH and N
21、LSThe experts would immediately recognise theBethe ansatz(BA)equations for the non-linear Schroedinger theory(NLS)Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.NLS=large spin limit of the SU(2)XXX spin chainYMH and NLSMoreover the NLS Hamiltoniansare the 0-obs
22、ervables of the theory,likeThe VEV of the observable=The eigenvalue of the HamiltonianQ ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.YMH and NLSSince 1997 nothing came out of this result.It could have been simply a coincidence.In 2006 A.Gerasimov and S.Shatashv
23、ili have revived the subjectHistoryYMH and interacting particlesGS 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 YM with the matter-fermions with pair-wise interaction HistoryMore importantl
24、y,GS suggested that TFT/QIS equivalence is much more universalTodayWe shall rederive the result of MNS from a modern perspectiveGeneralize to cover virtually all BA soluble systems both with finite and infinite spinSuggest natural extensions of the BA equationsHitchin equationsSolutions can be viewe
25、d as the susy field configurations for the N=2 gauged linear sigma model For adjoint-valued linear fieldsHitchin equationsThe moduli space MH of solutions is a hyperkahler manifoldThe integrals over MH are computed by the correlation functions of an N=2 d=2 susy gauge theoryHitchin equationsThe kahl
26、er form on MH comes fromtwisted tree level superpotentialThe epsilon-term comes from a twisted mass of the matter multiplet Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s picture.GeneralizationTake an N=2 d=2 gauge theory with matter,In some representation R of the gaug
27、e group GGeneralizationIntegrate out the matter fields,compute the effective(twisted)super-potentialon the Coulomb branchQ ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Mathematically speakingConsider the moduli space MR of R-Higgs pairswith gauge group G Up to
28、the action of the complexified gauge group GCMathematically speakingStability conditions:Up to the action of the compact gauge group GMathematically speakingPushforward the unit class down to the moduli space MG of GC-bundlesEquivariantly with respect to the actionof the global symmetry group K on M
29、R Mathematically speakingThe pushforward can be expressed in terms of the Donaldson-like classes of the moduli space MG 2-observables and 0-observablesMathematically speakingThe pushforward can be expressed in terms of the Donaldson-like classes of the moduli space MG 2-observables and 0-observables
30、Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Mathematically speakingThe masses are the equivariant parametersFor the global symmetry group K Q ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Vacua of the gauge theoryDue to
31、quantization of the gauge fluxQ ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Q ui ckTi m e?and aTI FF(U nco m pressed)decom pressorare needed to see thi s pi cture.For G=U(N)Vacua of the gauge theoryEquations familiar from yesterdays lectureQ ui ckTi m e?and aT
32、I FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Q ui ckTi m e?and aTI FF(U nco m pressed)decom pressorare needed to see thi s pi cture.For G=U(N)partitionsVacua of the gauge theoryFamiliar example:CPN model(N+1)chiral multiplet of charge+1Qi i=1,N+1U(1)gauge groupN+1 vacuumField co
33、ntent:Effective superpotential:Vacua of gauge theoryGauge group:G=U(N)Matter chiral multiplets:1 adjoint,mass fundamentals,massanti-fundamentals,mass Field content:Another example:Vacua of gauge theoryEffective superpotential:Vacua of gauge theoryEquations for vacua:Vacua of gauge theoryNon-anomalou
34、s case:Redefine:Vacua of gauge theoryVacua:Gauge theory-spin chainIdentical to the Bethe ansatz equations for spin XXX magnet:Gauge theory-spin chainVacua=eigenstates of the Hamiltonian:Table of dualitiesXXX spin chain SU(2)L spinsN excitationsU(N)d=2 N=2 Chiral multiplets:1 adjointL fundamentalsL a
35、nti-fund.Special masses!Table of dualities:mathematically speakingXXX spin chain SU(2)L spinsN excitations(Equivariant)Intersection theory on MR for Table of dualitiesXXZ spin chain SU(2)L spinsN excitationsU(N)d=3 N=1Compactified on a circle Chiral multiplets:1 adjointL fundamentalsL anti-fund.Tabl
36、e of dualities:mathematically speakingXXZ spin chain SU(2)L spinsN excitationsEquivariant K-theory of the moduli space MRTable of dualitiesXYZ spin chain SU(2),L=2N spinsN excitationsU(N)d=4 N=1Compactified on a 2-torus =elliptic curve E Chiral multiplets:1 adjointL=2N fundamentalsL=2N anti-fund.Mas
37、ses=wilson loops of the flavour group=points on the Jacobian of ETable of dualities:mathematically speakingXYZ spin chain SU(2),L=2N spinsN excitationsElliptic genus of the moduli space MRMasses=K bundle over E=points on the BunK of ETable of dualitiesIt is remarkable that the spin chain hasprecisel
38、y those generalizations:rational(XXX),trigonometric(XXZ)and elliptic(XYZ)that can be matched to the 2,3,and 4 dim cases.Algebraic Bethe AnsatzThe spin chain is solved algebraically using certain operators,Which obey exchange commutation relationsFaddeev et al.Faddeev-Zamolodchikov algebraAlgebraic B
39、ethe AnsatzThe eigenvectors,Bethe vectors,are obtained by applying these operators to the fake vacuum.ABA 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.ABA vs GAUGE THEORYHowever,if the th
40、eory is embedded into string theory via brane realization then changing N is easy:bring in an extra brane.Hanany-Hori02ABA vs GAUGE THEORYMathematically speaking We claim that the Algebraic Bethe Ansatz is most naturally related to the derived category of the category of coherent sheaves on some loc
41、al CYABA vs STRING THEORYTHUS:B is for BRANE!is for location!More general spin chainsThe SU(2)spin chain has generalizations to other groups and representations.I quote the corresponding Bethe ansatz equations from N.ReshetikhinGeneral groups/repsFor simply-laced group H of rank rGeneral groups/reps
42、For simply-laced group H of rank rLabel representations of the Yangian of H A.N.Kirillov-N.Reshetikhin modulesCartan matrix of HGeneral groups/repsfrom GAUGE THEORYTake the Dynkin diagram corresponding to H A simply-laced group of rank rQUIVER GAUGE THEORYSymmetriesQUIVER GAUGE THEORYSymmetriesQUIVE
43、R GAUGE THEORYCharged matterAdjoint chiral multipletFundamental chiral multipletAnti-fundamental chiral multipletBi-fundamental chiral multipletQUIVER GAUGE THEORYMatter fields:adjoints QUIVER GAUGE THEORYMatter fields:fundamentals+anti-fundamentals QUIVER GAUGE THEORYMatter fields:bi-fundamentals Q
44、UIVER GAUGE THEORYQuiver gauge theory:full content QUIVER GAUGE THEORY:MASSESAdjoints iQUIVER GAUGE THEORY:MASSESFundamentalsAnti-fundamentals ia=1,.,Li QUIVER GAUGE THEORY:MASSESBi-fundamentals ijQUIVER GAUGE THEORYWhat is so special about these masses?QUIVER GAUGE THEORYFrom the gauge theory point
45、 of view nothing special.QUIVER GAUGE THEORYThe mass puzzle!The mass puzzleThe Bethe ansatz-like equationsCan be written for an arbitrary matrix The mass puzzleHowever the Yangian symmetry Y(H)would get replaced by some ugly infinite-dimensional free algreba without nice representations The mass puz
46、zleTherefore 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 parametersAmong them are the fermion masses Is there a(hidden)symmetry principle behind them?The Standard Model has many free parameters In the supersymm
47、etric modelswe considered the mass tuning can be explained using a duality to some quantum integrable systemFurther generalizations:Superpotential from prepotentialTree level partInduced by twistFlux superpotential(Losev,NN,Shatashvili97)The N=2*theory on R2 X S2Superpotential from prepotentialMagne
48、tic fluxElectric fluxIn the limit of vanishing S2 the magnetic flux should vanishInstanton corrected BA equationsEffective S-matrix contains 2-body,3-body,interactionsInstanton corrected BA equationsQ ui ckTi m e?and aTI FF(U ncom pressed)decom pressorare needed to see thi s pi cture.Instanton corre
49、cted BA equationsThe prepotential of the low-energy effective theoryIs governed by a classical(holomorphic)integrable systemDonagi-Witten95Liouville tori=Jacobians of Seiberg-Witten curvesClassical integrable systemvsQuantum integrable systemThat system is quantized when the gauge theory is subject
50、tothe Omega-backgroundNN02NN,Okounkov03Braverman03Our quantum system is different!Blowing up the two-sphereWall-crossing phenomena(new states,new solutions)Something for the futureNaturalness of our quiversSomewhat unusual matter contentBranes at orbifolds typically lead to smth like Naturalness of
