The gauge theory - From Sigma Models to Four-dimensional QFT

Classical GeometryofQuantum IntegrabilityandGauge TheoryNikita NekrasovIHES

This is a work onexperimentaltheoretical physics•In collaboration withAlexei Rosly(ITEP)andSamson Shatashvili(HMI and Trinity College Dublin)

Earlier workG.Moore, NN, S.Shatashvili.,arXiv:hep-th/9712241;A.Gerasimov, S.Shatashvili.arXiv:0711.1472, arXiv:hep-th/0609024NN, S.Shatashvili,arXiv:0901.4744, arXiv:0901.4748, arXiv: 0908.4052NN, E.WittenarXiv:1002.0888Earlier work on instanton calculusA.Losev, NN, S.Shatashvili.,arXiv:hep-th/9711108, arXiv:hep-th/9911099;NN arXiv:hep-th/0206161;The papers ofN.Dorey, T.Hollowood , V.Khoze, M.Mattis,Earlier work on separated variables and D-branesA.Gorsky, NN, V.Roubtsov,arXiv:hep-th/9901089;

In the past few yearsa connectionbetween the followingtwo seemingly unrelatedsubjectswas found

The supersymmetricgauge theorieswith as little as 4 supersymmetrieson the one hand

The supersymmetric vacua ofthe (finite volume)gauge theoryare the stationary states ofa quantum integrable system

OperatorsThe « twisted chiral ring » operatorsO 1 , O 2 , O 3 , … , O nmap to the quantum HamiltoniansΗ 1 , Η 2 , Η 3 , … , Η n

The main ingredient of thecorrespondence:The effective twistedsuperpotential of the gaugetheory=The Yang-Yang function of thequantum integrable system

The effective twistedsuperpotential of the gaugetheory

The effective twisted superpotentialleads to the vacuum equations1

The effective twistedsuperpotentialIs a multi-valued function onthe Coulomb branch of the theory,depends on the parameters of the theory

The Yang-Yang functionof the quantum integrablesystemThe YY function was introducedby C.N.Yang and C.P.Yang in 1969For the non-linear Schroedinger problem.

The miracle of Bethe ansatz:The spectrum of the quantumsystem is described bya classical equation1

EXAMPLE:Many-body systemCalogero-Moser-Sutherland system

The elliptic Calogero-MosersystemN identical particles ona circle of radius βsubject to the two-body interactionelliptic potential

Quantummany-body systemsOne is interested in the β-periodic symmetric,L 2 -normalizable wavefunctions

It is clear that one shouldget aninfinite discrete energy spectrumβ

Many-body systemvsgauge theoryTheinfinite discrete spectrumofthe integrable many-body system=The vacua of the N=2 d=2 theory

The gauge theoryThe N=2 d=2 theory, obtainedby subjecting the N=2 d=4 theoryto an Ω-background in R 2Φ → Φ - ε D φN=2 d=2

The four dimensional gaugetheory on Σ x R 2 ,viewed SO(2) equivariantly,can be formally treated as aninfinite dimensional version of atwo dimensional gauge theory

The two dimensional theoryHas aneffective twistedSuperpotential!

The effective twisted superpotentialCan be computed from theN=2 d=4Instanton partition function

The effective twisted superpotentialΦ → Φ - ε 1 D φ1- ε 2 D φ2

The effective twisted superpotentialhas one-loop perturbativeand all-order instanton corrections

In particular, for the N=2 * theory(adjoint hypermultiplet with mass m)

N=2 * theory

Bethe equationsFactorized S-matrixThis is the two-body scatteringIn hyperbolic Calogero-Sutherland

Bethe equationsFactorized S-matrixTwo-bodypotential

Bethe equationsFactorized S-matrixHarish-Chandra, Gindikin-Karpelevich,Olshanetsky-Perelomov, Heckmann,final result: Opdam

The full superpotentialof N=2 * theory leads to the vacuumequationsMomentum phase shiftTwo-body scatteringThe finite size correctionsq = exp ( - Nβ )

Dictionary

DictionaryEllipticCMN=2* theorysystem

DictionaryclassicalEllipticCM4d N=2*theorysystem

DictionaryquantumEllipticCMsystem4d N=2*Theoryin 2dΩΩ-background

DictionaryThe(complexified)systemThe gaugecoupling τSize β

DictionaryThePlanckconstantTheEquivariantparameterε

DictionaryThe correspondenceExtends to otherintegrable sytems:Toda, relativistic Systems,Perhaps all 1+1 iQFTs

Two ways of gettingtwo dimensional theory startingwith a higher dimensional one1) Kaluza-Klein reduction, e.g.compactification on a toruswith twisted boundaryconditions…2) Boundary theory, localizationon a cosmic string….

Two ways of gettingtwo dimensional theory startingwith a higher dimensional one1) Kaluza-Klein reduction: givesthe spin chains, e.g. XYZ2) Boundary theory, localizationon a cosmic string: gives themany-body systems, e.g. CM,more generally, a Hitchinsystem

PlanNow let us followThe quantization proceduremore closely,Starting on the gauge theoryside

In the mid-nineties of the 20century it was understood,thatThe geometry of the moduli space ofvacua of N=2 supersymmetric gaugetheory is identified with that of a base ofan algebraic integrable systemDonagi, WittenGorsky, Krichever, Marshakov, Mironov, Morozov

Liouville toriThe Base

Special coordinateson the moduli spaceThe moduli spaceofvacua ofd=4 N=2 theory

The Coulomb branchof the moduli spaceof vacua of thed=4 N=2 supersymmetricgauge theoryis the baseof a complex (algebraic)integrable system

The Coulomb branchof the same theory,compactified on a circle downto three dimensionsis the phasespace ofthe same integrable system

This moduli space is ahyperkahler manifold, andit can arise both as a Coulombbranch of one susy gaugetheory and as a Higgs branchof another susy theory.This is the 3d mirrorsymmetry.

The moduli spaceofvacua ofd=3 N=4 theory

In particular, one can startwith asix dimensional (0,2)ADEsuperconformal field theory,and compactify it onΣ x S 1with the genus g Riemann surface Σ

The resulting effective susygauge theory in threedimensions will have 8supercharges (with theappropriate twist along Σ)

The resulting effective susygauge theory in threedimensions will have theHitchin moduli space as themoduli space of vacua.The gauge group in Hitchin’sequations will be the group ofthe same A,D,E typeas in the definition of the (0,2)theory.

The Hitchin moduli spaceis the Higgs branch of the5d gauge theorycompactified on Σ

The mirror theory, for which theHitchin moduli spaceis the Coulomb branch,is conjectured Gaiotto , in the A 1 case,to be the SU(2) 3g-3gauge theory in 4d,compactified on a circle,with some matter hypermultiplets inthetri-fundamental and/or adjointrepresentations

One can allow the Riemann surfacewith n punctures,with some local parametersassociated with the punctures.The gauge group is then SU(2) 3g-3+nwith matter hypermultiplets in thefundamental,bi-fundamental,tri-fundamental representations,and, sometimes, in the adjoint.

For example,the SU(2) with N f =4Corresponds to theRiemann surfaceof genus zerowith 4 punctures.The local data at the puncturesdetermines the masses

For example,the N=2* SU(2) theoryCorresponds to theRiemann surfaceof genus onewith 1 punctures.The local data at the puncturedetermines the mass of the adjoint

From now on we shall bediscussing these« generalized quivertheories »• The integrable system corresponding tothe moduli space of vacua of the 4dtheory is the SU(2) Hitchin system onThe punctured Riemann surface Σ

Hitchin systemGauge theory on a Riemann surfaceThe gauge field A µ and the twistedHiggsfield Φ µ in the adjointrepresentation are required to obey:

Hitchin equations

Hitchin systemModulo gauge transformations:We get the moduli space M H

Hyperkahler structure of M H• Three complex structures: I,J,K• Three Kahler forms: ω I , ω J , ω K• Three holomorphic symplectic forms:Ω I , Ω J , Ω K

Hyperkahler structure of M HThree Kahler forms: ω I , ω J , ω KThree holomorphic symplectic forms:Ω I = ω J + i ω K ,Ω J = ω K + i ω I ,Ω K = ω I + i ω J

Hyperkahler structure of M HA = A + i Φ

Hyperkahler structureThe linear combinations, parametrized by thepoints on a twistor two-sphere S 2a I + b J + c K, wherea 2 +b 2 +c 2 =1

The integrable structureIn the complex structure I,the holomorphic functions are: for each Beltramidifferential µ (i) , i=3g-3+n

The integrable structureThese functions Poisson-commute w.r.t. Ω I{ H i , H j } = 0

The integrable structureThe generalization to other groups is known,e.g. for G=SU(N)

The integrable structureThe action-angle variables:Fix the level of the integrals of motion,ie fix the values of all H i ’sEquivalently:fix the (spectral)curve C inside T*ΣDet( λ - Φ z ) = 0Its Jacobian is the Liouville torus, andThe periods of λdz give the special coordinatesa i , a Di

The quantum integrablestructureThe naïve quantization, using thatin the complex structure IM H is almost = T*MWhere M=Bun GΦ z becomes the derivativeH i become the differential operators.More precisely, one gets the space of twisted (by K 1/2 M)differential operators on M=Bun G

Thinking about the Ω - deformationof the four dimensional gauge theory,leads to the conclusion thatthe quantum Hitchin systemIs governed by a Yang-Yang function,The effective twisted superpotential

Here comes the experimentalfactThe effective twistedsuperpotential,the YY function ofthe quantum Hitchin system:In fact has aclassical mechanical meaning!

M H as themoduli space of G C flatconnectionsIn the complex structure J theholomorphic variables are:A µ = A µ + i Φ µwhich obey (modulo complexified gauge transformations):F= dA+ [A,A] = 0

M H as themoduli space of G C flatconnectionsIn this complex structure M His defined without a reference to thecomplex structure of ΣM H = Hom ( π 1 (Σ) , G C ) / G C

M H as themoduli space of G C flatconnectionsHowever M HContains interesting complex Lagrangiansubmanifolds which do depend on thecomplex structure of ΣL Σ= the variety of G-opersBeilinson, DrinfeldDrinfeld, Sokolov

M H as themoduli space of G C flatconnectionsL Σ= the variety of G-opersBeilinson, DrinfeldDrinfeld, SokolovThe Beltrami differentialµ is fixed, the projectivestructure T is arbitrary,provided

M H as themoduli space of G C flatconnectionsL Σ= the variety of G-opersThe Beltrami differential µ is fixed, the projective structure Tis arbitrary, provided it is compatible with the complexstructure defined by

M H as themoduli space of G C flatconnectionsL Σ =i.e.

Opers on a sphereFor example, on a two-sphere with n punctures, theseconditions translate to the following definition of the spaceof opers with regular singularities: we are studying thespace of differential operators of second order, of the form

Opers on a sphereWhere Δ i are fixed,while the accessory parameters ε i obey

Opers on a sphereAll in all we get a (n-3)-dimensional subvariety in the2(n-3) dimensional moduli space of flat connections on then-punctured sphere with fixed conjugacy classes of themonodromies around the punctures:m i = Tr (g i )

The main conjectureThe YY function is thegenerating function ofthe variety of opers

The variety of opers isLagrangian withrespect to Ω J

We shall now constructa system of Darbouxcoordinates on M H

α i , β i

So thatL Σ= the variety of G-opers,is described by the equations

The moduli space is going to be covered by amultitude of Darboux coordinate charts, one perevery pair-of-pants decomposition(and some additional discrete choice)

Equivalently,a coordinate chartU Γper maximal degenerationΓ of the complex structure on Σ

The maximal complex structure degenerations =The weakly coupled gauge theory descriptions ofGaiotto theories, e.g. for the previous example

The α i coordinates are nothing but the logarithmsof the eigenvalues of the monodromies aroundthe blue cycles:

The α i coordinates are nothing but the logarithmsof the eigenvalues of the monodromies aroundthe blue cycles:cf. Drukker, Gomis, Okuda, Teschner;Verlinde; Verlinde

The β i coordinates are defined from the local datainvolving the cycle C i and itsfour neighboring cycles(or one, if the blue cycle belongs to a genus onecomponent) :

The local data involving the cycle C i andits four neighboring cycles :

C i

The local picture:four holes, two interesting holonomiesHol C ~ g 2 g 1Hol Cv~ g 3 g 2

Complexified hyperbolic geometry:

The coordinates α i ,β i can bethus explicitly expressed in terms ofthe traces of the monodromies:B = Tr (Hol Cv~ g 3 g 2 )= Tr g i= Tr (Hol C ~ g 2 g 1 )

The construction of the hyperbolic polygongeneralizes to the case of n punctures:

The construction of the hyperbolic polygongeneralizes to the case of n punctures:

For g i obeying some reality conditions,e.g. SU(2), SL(2,R), SU(1,1), SO(1,2), or, R 3We get the real polygons inS 3 , H 3 , R 2,1 , E 3our coordinates reduce tothe ones studied byKlyachko,Kapovich, MillsonKirwan, Foth,NB: The Loop quantum gravitycommunity (Baez, Charles, Rovelli,Roberts, Freidel, Krasnov, Livin, …. )uses different coordinates

Our polygons sit in the group manifoldAn interesting problem:Relate our coordinates to thecoordinatesIntroduced by Fock and Goncharov,Based on triangulations of theRiemann surface with punctures.

Our polygons sit in the group manifoldAn interesting problem:Relate our coordinates to thecoordinatesIntroduced by Fock and Goncharov,Based on triangulations of theRiemann surface with punctures.The FG coordinates are the basis ofthe Gaiotto-Moore-Neitzkework on the hyperkahler metric onM H

The local data involving the cycle C i onthe genus one componentg 2g 1

The canonical transformations(the patching of thecoordinates)sut

The s-t channel flop:the generating function is thehyperbolic volumest

The s-t channel flop:the generating function is thehyperbolic volumeV v is the volume of the dual tetrahedron

The s-u flop =composition ofthe 1-2 exchange(a braid group action)and the flopThe 1-2 braiding acts as:(α,β) goes to(α, β ± α + πi)

The theoryWhy did the twisted superpotential turn into a generating function?Why did the variety of opers showed up?What is the meaning of Bethe equations forquantum Hitchin in termsof this classical symplectic geometry?Why did these hyperbolic coordinates(which generalize the Fenchel-Nielsen coordinates onTeichmuller space and Goldman coordinates on the moduliof SU(2) flat connections) become the special coordinates in thetwo dimensional N=2 gauge theory?What is the relevance of the geometry of hyperbolic polygons forthe M5 brane theory?For the three dimensional gravity?For the loop quantum gravity?

The theoryWhy did the twisted superpotential turn into a generating function,and why did the variety of opers showed up?This can be understood by viewing the 4d gauge theory as a 2dtheory with an infinite number of fields in two different ways(NN+EW)

The theoryWhat is the meaning of Bethe equations forquantum Hitchin in termsof this classical symplectic geometry?They seem to describe an intersection of the brane of opers withanother (A,B,A) brane, a more conventional Lagrangian brane.The key seems to be in the Sklyanin’s separation of variables(NSR, in progress)

The theoryWhy did these hyperbolic coordinates(which generalize the Fenchel-Nielsen coordinates onTeichmuller space and Goldman coordinates on the moduliof SU(2) flat connections) become the special coordinates in thetwo dimensional N=2 gauge theory?The key seems to be in the relation tothe Liouville theory and the SL(2,C) Chern-Simons theory.A concrete prediction of our formalism is the quasiclassical limit ofthe Liouville conformal blocks:

The quasiclassical limit of the Liouville conformal blocks(motivated by the AGT conjecture,but it is independent of the validity of the AGT):

In the genus zero case it should imply the Polyakov’s conjecture(proven for Fuchsian m’s by Takhtajan and Zograf);can be compared with the results of Zamolodchikov,Zamolodchikov; Dorn-Otto

The theory vs experimentThe conjecture in gauge theoryhas been tested to a few orders ininstanton expansion for simplest theories (g=0,1),and at the perturbative level of gauge theory for all theories.What is lacking is a good understanding of the theories with trifundamentalhypermultiplets (in progress, NN+V.Pestun)

The prediction of the theoryThe conjecture implies that theTwisted superpotential transforms underthe S-duality in the following way:a generalization of thefour dimensional electric-magnetic transformation of theprepotential

For the rest of thepuzzlesthere remains much tobe said,Hopefully in the nearfuture.Thank you!