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

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1、,Two Dimensional Gauge Theoriesand Quantum Integrable Systems,Nikita Nekrasov IHESImperial College April 10,2008,Based on,NN,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.Shat

2、ashvili 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,And quantum integrable systemssoluble by Bethe Ansatz techniques.,Mathematically speaking,the cohomology,K-theory and elliptic cohomology of various

3、 gauge theory moduli spaces,like moduli of flat connections and instantons,And 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-)History,In 1992 E.Witten studied two dimensional Yang-Mills

4、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

5、 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-con

6、nections=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 t

7、heories 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 li

8、mit,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-Millstheories define topological field theories(TFT),Yang-Mills theories in 2d,Both physical and co

9、homological Yang-Millstheories 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,Gro

10、ssDouglas,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 defor

11、med 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 loopin some representation V of the gauge group:,YM in 2d and particles on a circle,One gets Calogero-Sutherland(spin)particles on a cir

12、cle(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 v

13、arious 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 theory,Inspired by the work of H.Nakajima,Yang-Mills-Higgs theory,Among various examples,MNS studied H

14、itchins moduli space MH,Yang-Mills-Higgs theory,Unlike 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 volume is infinite.,Yang-Mills-Higgs theory,In order to cure this infnity in a re

15、asonable 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 equation

16、s:,YMH and 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 NLS,Moreover the NLS Hamiltoniansare the 0-observables of the theory,like,The VEV of the observable=The eigenvalue o

17、f 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 p

18、articles 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 perspectiveGeneralize t

19、o cover virtually all BA soluble systems both with finite and infinite spinSuggest 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 spa

20、ce MH of solutions is a hyperkahler manifoldThe 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 fromtwisted tree level superpotentialThe epsilon-term comes from a twisted mass of the matter multiplet,Generaliza

21、tion,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-potentialon the Coulomb branch,Mathematically speaking,Consider the moduli space MR of R-Higgs pairswith gauge group G,Up to

22、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-bundlesEquivariantly with respect to the actionof the global symmetry group K

23、 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-obs

24、ervables,Mathematically speaking,The masses are the equivariant parametersFor 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,Fa

25、miliar example:CPN model,(N+1)chiral multiplet of charge+1Qi i=1,N+1U(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,massanti-fundamentals,mass,Field content:,Another example:,Vacua of gau

26、ge 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 Hamilt

27、onian:,Table of dualities,XXX spin chain SU(2)L spinsN excitations,U(N)d=2 N=2 Chiral multiplets:1 adjointL fundamentalsL anti-fund.,Special masses!,Table of dualities:mathematically speaking,XXX spin chain SU(2)L spinsN excitations,(Equivariant)Intersection theory on MR for,Table of dualities,XXZ s

28、pin chain SU(2)L spinsN excitations,U(N)d=3 N=1Compactified on a circle Chiral multiplets:1 adjointL fundamentalsL anti-fund.,Table of dualities:mathematically speaking,XXZ spin chain SU(2)L spinsN excitations,Equivariant K-theory of the moduli space MR,Table of dualities,XYZ spin chain SU(2),L=2N s

29、pinsN excitations,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,Table of dualities:mathematically speaking,XYZ spin chain SU(2),L=2N spinsN excitations,Elliptic ge

30、nus 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 hasprecisely 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

31、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 natur

32、al to fix L=total number of spinsand 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,Mathemat

33、ically 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

34、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,Genera

35、l groups/repsfrom 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 THEORYCharged matter,Adjoint chiral multiplet,Fundamental chiral multiplet,Anti-fundamental chiral multiplet,Bi-fundamen

36、tal 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,Fun

37、damentalsAnti-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 equation

38、s,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

39、 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 modelswe considered the mass tuning can be explained using a duality to some quantum integrabl

40、e 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

41、 corrected BA equations,Effective S-matrix contains 2-body,3-body,interactions,Instanton corrected BA equations,Instanton corrected BA equations,The prepotential of the low-energy effective theoryIs governed by a classical(holomorphic)integrable system,Donagi-Witten95,Liouville tori=Jacobians of Sei

42、berg-Witten curves,Classical integrable systemvsQuantum integrable system,That system is quantized when the gauge theory is subject tothe Omega-background,NN02NN,Okounkov03Braverman03,Our quantum system is different!,Blowing up the two-sphere,Wall-crossing phenomena(new states,new solutions),Somethi

43、ng for the future,Naturalness of our quivers,Somewhat unusual matter contentBranes at orbifolds typically lead to smth like,Naturalness of our quivers,This picture would arise in the sa(i)0 limit,BA for QCDFaddeev-Korchemsky94,Naturalness of our quivers,Other quivers?,Naturalness of our quivers,Poss

44、ibly with the help of K.Saitos construction,CONCLUSIONS,We found the Bethe Ansatz equations are the equations describing the vacuum configurations of certain quiver gauge theories in two dimensionsThe 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 theories4.Obviously this is a beginning of a beautiful story.,