Quantum q-Langlands Correspondence 7 1 0 2 Mina Aganagic1,2, Edward Frenkel2, Andrei Okounkov3 n a J 1Center for Theoretical Physics, University of California, Berkeley 1 1 2Department of Mathematics, University of California, Berkeley ] h 3Department of Mathematics, Columbia University t - p e h Abstract [ 1 We formulate a two-parameter generalization of the geometric Langlands correspon- v 6 dence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks 4 1 of the quantum affine algebra and the deformed W-algebra associated to two Langlands 3 0 dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima . 1 quiver varieties. The physical origin of the correspondence is the 6d little string theory. 0 7 The quantum Langlands correspondence emerges in the limit in which the 6d string theory 1 : v becomes the 6d conformal field theory with (2,0) supersymmetry. i X r a Contents 1 Introduction 3 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Statement of the correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Geometry behind the correspondence . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Main steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Non-simply laced groups and folding . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 String theory origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 q-deformed conformal blocks 19 2.1 Electric side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Magnetic side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Conformal limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Integral representation of vertex functions 26 3.1 Quasimaps to Nakajima varieties . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Vertex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Localization contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Integral formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Vertex functions and W algebra blocks. . . . . . . . . . . . . . . . . . . . . 39 q,t 4 Vertex functions and qKZ 42 4.1 Degeneration formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 g=A Example 51 1 5.1 qKZ equation and its z-solutions . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Geometric interpretation in terms of X =T∗Pn−1 . . . . . . . . . . . . . . . 53 5.3 q-Virasoro conformal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4 Elliptic stable envelope and z- and a-solutions . . . . . . . . . . . . . . . . . . 56 5.5 X =T∗P1 example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1 5.6 Conformal limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Connection with the geometric Langlands Correspondence 61 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.3 Oneness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.4 A proposal for quantum Langlands . . . . . . . . . . . . . . . . . . . . . . . . 66 6.5 Connection with Lurie’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . 69 6.6 Conformal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.7 Integral representation of conformal blocks. . . . . . . . . . . . . . . . . . . . 70 6.8 Explicit identification of conformal blocks . . . . . . . . . . . . . . . . . . . . 75 7 Quivers from String Theory 79 7.1 3d quiver gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.2 Quiver gauge theory from IIB string . . . . . . . . . . . . . . . . . . . . . . . 80 7.3 Little string theory from IIB string . . . . . . . . . . . . . . . . . . . . . . . . 81 7.4 Non-simply laced case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.5 Conformal limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8 Vertex Functions from Physics 86 8.1 Little string partition function . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.2 Localization to defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.3 Defect partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.4 Index for non-simply laced g. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.5 Vertex functions from 3d gauge theory . . . . . . . . . . . . . . . . . . . . . . 89 8.6 Conformal limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9 Langlands Correspondence From Little Strings 96 9.1 S duality of 4d Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . 96 9.2 Derivation of S-duality from little string theory . . . . . . . . . . . . . . . . . 97 9.3 Gauge theory partition function from little string . . . . . . . . . . . . . . . . 99 9.4 Little string defects and line operators in gauge theory . . . . . . . . . . . . . 101 A Integral formulas in K-theory of GIT quotients 102 2 1 Introduction 1.1 Overview In the 50 years of its existence, the Langlands program and the Langlands philosophy have grown to encompass many objects of central importance to both mathematics and mathematical physics. In particular, the geometric Langlands correspondence starts with a complex projective algebraic curve C with the goal, as it is usually understood today, to prove an equivalence betweencertaincategoriesassociatedtoapairG,LGofLanglandsdualconnectedreductive complex Lie groups. These are certain categories of sheaves (of D-modules and O-modules, respectively) on the moduli stack Bun of LG-bundles on C and the moduli stack Loc of LG G flat G-bundles on C.1 Kapustin and Witten have shown [63] that this equivalence is closely relatedtoS-dualityofmaximallysupersymmetric4dgaugetheorieswithgaugegroupsbeing the compact forms of G and LG. Beilinson and Drinfeld have constructed in [15] an important part of the geometric Langlands correspondence using the isomorphism [34] between the center of the (chiral) affine Kac–Moody algebra L(cid:99)g at the critical level Lk = −Lh∨ and the classical W-algebra W (g). Their construction is closely connected to the 2d conformal field theory and the ∞ theory of chiral (or vertex) algebras (see [42] for a survey; and also [116] in which an analogy between 2d CFT and the theory of automorphic representations was first observed and investigated). SincethelevelofL(cid:99)gmaybedeformedawayfromthecriticalvalue,andatthesametime W (g) may be deformed to the quantum W-algebra W (g), one is naturally led to look for ∞ β a quantum deformation of the geometric Langlands correspondence. Though many interesting structures have emerged in the studies under the umbrella of “quantum geometric Langlands” (from the point of view of 2d CFT [39, 40, 105, 106, 112, 53, 54]; in the framework of 4d gauge theory [63, 62, 45]; and, in the abelian case, as a deformation of the Fourier–Mukai transform [96]), it seems that at the moment there is no canonical definition of “quantum geometric Langlands correspondence” in the literature in the non-abelian case. 1Theexistenceofsuchanequivalence,whichmaybeviewedasacategoricalnon-abelianFouriertrans- form,wasoriginallyproposedbyBeilinsonandDrinfeld;later,apreciseconjecturewasformulatedin[10]. We note that some of our notation bucks the usual conventions. In particular, the roles of G and LG are exchanged. 3 1.1.1 For us, the main feature of the quantum geometric Langlands correspondence is an iso- morphism between the spaces of conformal blocks of certain representations of two chiral algebras: L(cid:99)g ←→ W (g), (1.1) Lk β the affine Kac–Moody algebra of Lg at level Lk and the W-algebra W (g). The algebra β W (g) is obtained by the quantum Drinfeld–Sokolov reduction [33, 17, 34] of the affine β algebra g at level k, where β =m(k+h∨) in the notation of [48].2 (cid:98) Wewillestablishthisisomorphismandproveastrongerresultinthecaseofsimply-laced g: an identification of conformal blocks of the two algebras if the parameters are generic and related by the formula 1 β−m= . (1.2) L(k+h∨) Therelationbetweenthecorrespondingchiralalgebrasmaybeviewedasastrong/weak coupling transformation. Indeed, if we define τ = β/m and Lτ = −L(k+h∨), then (1.2) says that τ −1=−1/(mLτ), (1.3) and so Lτ near zero corresponds to large values of τ. The parameters τ and Lτ are related to the complexified coupling constants of the two S-dual 4d Yang-Mills theories. Note the shiftτ (cid:55)→τ−1,ascomparedtotheW-algebradualityformulaof[34](seeSection6formore details). This is a shift of the theta angle from 4d gauge theory perspective (see Section 9). The identification of these conformal blocks emerges in a canonical fashion, but only after we introduce one more parameter and perform one more deformation. 1.1.2 We consider a two-parameter deformation of the geometric Langlands correspondence: the q-deformation together with the deformation away from the critical level. This turns out to be a productive point of view. Namely, we replace the above chiral algebras with their deformed counterparts: the quantumaffinealgebraU(cid:126)(L(cid:99)g),whichisan(cid:126)-deformationoftheuniversalenvelopingalgebra of L(cid:99)g introduced by Drinfeld and Jimbo [29, 61], and the deformed W-algebra W (g) q,t 2Thus,whatwedenoteherebyW (g)isW (g)of[44,42],whereβ=m(k+h∨). Inourpresentnotation, β k theclassicalW-algebraassociatedtogisW∞(g). SeeSection6formoredetails. 4 introduced in [48] (see also [103, 36, 11] for g=sl ), which is a deformation of W (g). We n β will refer to both of these as “q-deformations”, both for brevity and because q will appear as a step in difference equations that are of principal importance to us. (In our notation, thequantumaffinealgebraU(cid:126)(L(cid:99)g)becomestheenvelopingalgebraofL(cid:99)ginthelimit(cid:126)→1; this agrees with the notation used in [91]. For a fixed non-critical value of Lk, this limit is the same as the limit q →1.) We focus on the case that the curve C is an infinite cylinder, C ∼=C× ∼=infinite cylinder. It should be noted that integrable deformations away from the conformal point are unlikely to exist unless C is flat. The torus case should follow from the case of the cylinder, by imposing periodic identifications.3 The case when C is a plane can be obtained from ours, by taking the radius of the cylinder to infinity. Weconjecture(andproveinthesimply-lacedcase)acorrespondencebetweenq-deformed conformal blocks of the quantum affine algebra U(cid:126)(L(cid:99)g) and the deformed W-algebra U(cid:126)(L(cid:99)g) ←→ Wq,t(g), (1.4) where the parameters q =(cid:126)−L(k+h∨), t=qβ, (1.5) are generic and related by the formula t=qm/(cid:126) (1.6) which yields (1.3). It is this identification of the deformed conformal blocks that we refer to as a “quantum q-Langlands correspondence” in the title of the present paper. Thephysicalsettingforthecorrespondenceisasix-dimensionalstringtheory, calledthe “(2,0)littlestringtheory”. Thelittlestringtheory[102,74]isaone-parameterdeformation of the ubiquitous 6d (2,0) superconformal theory (see e.g. [119]). The deformation corre- sponds to giving strings a non-zero size, and “converts” the relevant chiral algebras, such as g and W (g), into the corresponding deformed algebras. (cid:98) β 3TogetthedeformedconformalblocksonatorusC=C×/pZ,onewouldstudywithblocksonC=C×, butwithinsertionsthatareinvariantundertheactionofpZ. 5 1.1.3 Some preliminary remarks about deformed conformal blocks are in order. In the case of an affine Kac–Moody algebra and a cylinder C, the space of conformal blocks is isomorphic to the space of solutions of the Kniznik-Zamolodchikov (KZ) equations, which behave well as the insertion points are taken to infinity. The space of q-deformed conformal blocks for quantum affine algebras can be similarly defined, following [49], as the space of solutions of the quantum Kniznik-Zamolodchikov (qKZ) equations. In either case, there is a particular fundamental solution of the equations which comes from sewing chiral vertex operators. This solution is given by (1.8) in the case of deformed conformal blocks of U(cid:126)(L(cid:99)g). To the best of our knowledge, the definition of the space of deformed conformal blocks forthedeformedW-algebraW (g)wasnotavailableintheliteratureuntilnow. Theblocks q,t formally equal correlation functions of free field vertex operators of the deformed W (g) q,t algebra in (1.10), constructed in [48], however the definition is not complete. (One has yet to specify the space of allowed contours of integration for screening charges.) Moreover, the analogues of the qKZ equations were previously unknown for the deformed W-algebras W (g), as far as we know. q,t Inthispaper,wegiveadefinitionofdeformedconformalblocksofW (g)forthecylinder q,t C as vertex functions a Nakajima quiver variety X, whose quiver diagram is based on the Dynkindiagramofg. ThevertexfunctionsareequivariantK-theoreticcountsofquasimaps C (cid:57)(cid:57)(cid:75) X, of all possible degrees. We will prove that this definition is compatible with the free field integral representation (1.10), for a suitable choice of integration contours. With this definition, we prove that there is an identification of the deformed confor- mal blocks of U(cid:126)(L(cid:99)g) in (1.8) with the deformed conformal blocks of Wq,t(g) in (1.10), for simply-laced g, provided that the parameters of the two algebras are generic related by equation (1.5). The identification of the U(cid:126)(L(cid:99)g)-conformal blocks from (1.8) with the W (g)-conformal blocks has the following general form: q,t specific covector× U(cid:126)(L(cid:99)g) conformal blocks= (1.7) W (g) algebra blocks ×elliptic stable envelopes q,t The elliptic stable basis is a basis of equivariant elliptic cohomology of X defined and constructedin[4]. Thespecialcovectorin(1.7)isaWhittakertypevector; itistheidentity vector in the K-theoretic stable basis of X. Conversely, we show that vertex functions of X with descendent insertions at 0 ∈ C, expressed in the elliptic stable basis of X, lead 6 to deformed conformal blocks of U(cid:126)(L(cid:99)g) given by (1.8). This leads to Mellin-Barnes type integral formulas for U(cid:126)(L(cid:99)g) conformal blocks for Lg simply laced, which are new, as far as we know. We also provide the previously missing difference equations satisfied by the deformed conformalblocksofW (g). Itfollowsfrom(1.7)thatthesearescalardifferenceequivalents q,t of the qKZ equations satisfied by the corresponding U(cid:126)(L(cid:99)g) conformal blocks. The identifi- cation between the spaces of deformed conformal blocks of U(cid:126)(L(cid:99)g) and Wq,t(g) then follows as a corollary. From the physics perspective, vertex functions and the elliptic stable envelopes are par- tition functions of the three-dimensional (3d) supersymmetric quiver gauge theory whose Higgs branch is the Nakajima variety X. The 3d gauge theories are theories on defects of littlestringtheory,whichleadtovertexoperatorsin(1.8)and(1.10). Supersymmetryleads to localization: the partition function of the 6d string theory with defects is equal to the partition function of the theory on the defects (see Section 8). 1.2 Statement of the correspondence Let x be a coordinate on C ∼= C×. Fix a finite collection of distinct points on C, with coordinates a . We propose, and prove in the simply-laced case, a correspondence between i the following two types of q-conformal blocks on C. 1.2.1 On the electric side, we consider the quantum affine algebra U(cid:126)(L(cid:99)g) blocks [49] (cid:89) (cid:104)λ(cid:48)| Φ (a )|λ(cid:105) (1.8) Lρi i i whereΦLρ(x)isachiralvertexoperatorcorrespondingtoafinite-dimensionalU(cid:126)(L(cid:99)g)-module Lρ. Thestate|λ(cid:105)isthehighestweightvectorinalevelLkVermamodule. Itsweightλ∈Lh∗ is an element of the dual of the Cartan subalgebra for Lg. This is illustrated in Figure 1. Itsufficestofocusonvertexoperatorscorrespondingtothefundamentalrepresentations because all others may be generated from these, by fusion. The highest weight of a funda- mental representation is one of the fundamental weights Lw of Lg. The conformal block a (1.8) takes values in a weight subspace of ⊗ (Lρ )=⊗ (Lρ )⊗ma, i i a a 7 Figure 1: The cylinder C with the insertions of vertex operators corresponding to finite-dimensional U(cid:126)(L(cid:99)g)-modules Lρi at the points ai ∈C. Boundary conditions at infinity are the highest weight vectors (cid:104)λ(cid:48)| and |λ(cid:105). namely, it has weight=λ(cid:48)−λ (cid:88) (cid:88) = m Lw − d Le , d ≥0. (1.9) a a a a a a a In (1.9), we write the weight as the difference of the highest weights and simple positive roots Le of Lg. The index a runs here from 1 to rk(g). a 1.2.2 On the magnetic side, we consider q-correlators of the W (g) algebra of the form q,t (cid:104)µ(cid:48)| (cid:89)V∨(a )(cid:89)(cid:16)Q∨(cid:17)da |µ(cid:105). (1.10) i i a i a V∨(x) and Q∨ are the vertex and the screening charge operators defined by E. Frenkel a a and N. Reshetikhin in [48]. They are labeled by coroots and coweights of g, respectively. Recall that Langlands duality maps coweights and coroots of g to weights and roots of Lg, respectively. The screening charge operators are defined as integrals of screening current vertex operators Q∨ = (cid:82) dx S∨(x), so (1.10) is implicitly an integral formula for W (g) a a q,t algebra blocks. The coweights of g labeling V∨(x) are the highest weights of the fundamental repre- a sentations of Lg. The operator V∨(a ), inserted at a point on C with the coordinate a , is i i i associated to the same representation of Lg as the corresponding vertex operator in (1.8). The state |µ(cid:105), labeled by an element µ ∈ h of the Cartan subalgebra of g, generates an irreducible Fock representation of the W (g) algebra [48]. The (co)weights µ and µ(cid:48) are q,t determined by λ and λ(cid:48) (the exact formula depends on the chosen normalization). 8 1.3 Geometry behind the correspondence The central ingredient of our proof is that for Lie algebras of simply-laced type, when Lg=g, we can realize the q-conformal blocks (1.8) and (1.10) as vertex functions in equivariant quantum K-theory of a certain holomorphic symplectic variety X. The variety X is the Nakajima quiver variety with quiver Q=Dynkin diagram of g. 1.3.1 A Nakajima quiver variety X is a hyper-K¨ahler quotient (or a holomorphic symplectic reduction) X =T∗RepQ////G , (1.11) Q where RepQ=⊕ Hom(V ,V )⊕ Hom(V ,W ) (1.12) a→b a b a a a and (cid:89) (cid:89) G = GL(V ), G = GL(W ). (1.13) Q a W a a a The arrows in (1.12) are the arrows of the quiver. The dimensions of the vector spaces V a and W correspond as follows a dimV =d , dimW =m a a a a to the weight space data in (1.9). 1.3.2 Thequotientin(1.11)involvesageometricinvarianttheory(GIT)quotient, whichdepends on a choice of stability conditions. As a result, vertex functions also depend on a stability condition. This stability condition makes them analytic in a certain region of the K¨ahler moduli space of X. The transition matrix between vertex functions and U(cid:126)(L(cid:99)g) confor- mal blocks (1.8) will similarly depend on the stability condition. This dependence will be understood in what follows. 9