REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE

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1 REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE TOMOYUKI ARAKAWA Abstract. We survey a number of results regarding the representation theory of W -algebras and their connection with the resent development of the four dimensional N = 2 superconformal field theories. 1. Introduction (Affine) W -algebras appeared in 80 s in the study of the two-dimensional conformal field theory in physics. They can be regarded as a generalization of infinitedimensional Lie algebras such as affine Kac-Moody algebras and the Virasoro algebra, although W -algebras are not Lie algebras but vertex algebras in general. W - algebras may be also considered as an affinizaton of finite W -algebras introduced by Premet ([Pre02]) as a natural quantization of Slodowy slices. W -algebras play important roles not only in conformal field theories but also in integrable systems (e.g. [DS84, DSKV13, BM13]), the geometric Langlands program (e.g. [Fre07, AFO17]) and four-dimensional gauge theories (e.g. [AGT10, SV13, MO12, BFN16a]). In this note we survey the resent development of the representation theory of W -algebras. One of the fundamental problems in W -algebras was the Frenkel- Kac-Wakimoto conjecture [FKW92] that stated the existence and construction of rational W -algebras, which generalizes the integrable representations of affine Kac- Moody algebras and the minimal series representations of the Virasoro algebra. The notion of the associated varieties of vertex algebras played a crucial role in the proof of the Frenkel-Kac-Wakimoto conjecture, and has revealed an unexpected connection of vertex algebras with the geometric invariants called the Higgs branches in the four dimensional N = 2 superconformal field theories. Acknowledgments. The author benefited greatly from discussion with Christopher Beem, Davide Gaiotto, Madalena Lemos, Victor Kac, Anne Moreau, Hiraku Nakajima, Takahiro Nishinaka, Wolfger Peelaers, Leonardo Rastelli, Shu-Heng Shao, Yuji Tachikawa, and Dan Xie. The author is partially supported in part by JSPS KAKENHI Grant Numbers 17H01086, 17K Vertex algebras A vertex algebra consists of a vector space V with a distinguished vacuum vector 0 V and a vertex operation, which is a linear map V V V ((z)), written a b a(z)b = ( n Z a (n)z n 1 )b, such that the following are satisfied: (Unit axioms) ( 0 )(z) = 1 V and a(z) 0 a + zv [[z]] for all a V. (Locality) (z w) n [a(z), b(w)] = 0 for a sufficiently large n for all a, b V. 1

2 2 TOMOYUKI ARAKAWA The operator T : a a ( 2) 0 is called the translation operator and it satisfies (T a)(z) = [T, a(z)] = z a(z). The operators a (n) are called modes. For elements a, b of a vertex algebra V we have the following Borcherds identity for any m, n Z: [a (m), b (n) ] = ( ) m (1) (a j (j) b) (m+n j), j 0 (a (m) b) (n) = ( ) ( 1) j m (2) (a j (m j) b (n+j) ( 1) m b (m+n j) a (j) ). j 0 By regarding the Borcherds identity as fundamental relations, representations of a vertex algebra are naturally defined (see [Kac98, FBZ04] for the details). One of the basic examples of vertex algebras are universal affine vertex algebras. Let G be a simply connected simple algebraic group, g = Lie(G). Let ĝ = g[t, t 1 ] CK be the affine Kac-Moody algebra associated with g. The commutation relations of ĝ are given by (3) [xt m, yt n ] = [x, y]t m+n + mδ m+n,0 (x y)k, [K, ĝ] = 0 (x, y g, m, n Z), where ( ) is the normalized invariant inner product of g, that is, ( ) = 1/2h Killing form and h is the dual Coxeter number of g. For k C, let V k (g) = U(ĝ) U(g[t] CK) C k, where C k is the one-dimensional representation of g[t] CK on which g[t] acts trivially and K acts as multiplication by k. There is a unique vertex algebra structure on V k (g) such that 0 = 1 1 is the vacuum vector and x(z) = n Z(xt n )z n 1 (x g). Here on the left-hand-side g is considered as a subspace of V k (g) by the embedding g V k (g), x (xt 1 ) 0. V k (g) is called the universal affine vertex algebra associated with g at a level k. The Borcherds identity (1) for x, y g V k (g) is identical to the commutation relation (3) with K = k id, and hence, any V k (g)-module is a ĝ-module of level k. Conversely, any smooth ĝ-module of level k is naturally a V k (g)-module, and therefore, the category V k (g) -Mod of V k (g)-modules is the same as the category of smooth ĝ-modules of level k. Let L k (g) be the unique simple graded quotient of V k (g), which is isomorphic to the irreducible highest weight representation L(kΛ 0 ) with highest weight kλ 0 as a ĝ-module. The vertex algebra L k (g) is called the simple affine vertex algebra associated with g at level k, and L k (g) -Mod forms a full subcategory of V k (g) -Mod, the category of smooth ĝ-modules of level k. A vertex algebra V is called commutative if both sides of (1) are zero for all a, b V, m, n Z. If this is the case, V can be regarded as a differential algebra (=a unital commutative algebra with a derivation) by the multiplication a.b = a ( 1) b and the derivation T. Conversely, any differential algebra can be naturally equipped with the structure of a commutative vertex algebra. Hence, commutative vertex algebras are the same 1 as differential algebras ([Bor86]). 1 However, the modules of a commutative vertex algebra are not the same as the modules as a differential algebra.

3 REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 3 Let X be an affine scheme, J X the arc space of X that is defined by the functor of points Hom(Spec R, J X) = Hom(Spec R[[t]], X). The ring C[J X] is naturally a differential algebra, and hence is a commutative vertex algebra. In the case that X is a Poisson scheme C[J X] has the structure of Poisson vertex algebra ([A12a]), which is a vertex algebra analogue of Poisson algebra (see [FBZ04, Kac15] for the precise definition). It is known by Haisheng Li [Li05] that any vertex algebra V is canonically filtered, and hence can be regarded 2 as a quantization of the associated graded Poisson vertex algebra gr V = p F p V/F p+1 V, where F V is the canonical filtration of V. By definition, F p V = span C {(a 1 ) ( n1 1)... (a r ) ( nr 1) 0 a i V, n i 0, n i p}. i The subspace R V := V/F 1 V = F 0 V/F 1 V gr V is called Zhu s C 2 -algebra of V. The Poisson vertex algebra structure of gr V restricts to the Poisson algebra structure of R V, which is given by The Poisson variety ā. b = a ( 1) b, {ā, b} = a (0) b. X V = Specm(R V ) called the associated variety of V ([A12a]). We have [Li05] the inclusion (4) Specm(gr V ) J X V. A vertex algebra V is called finitely strongly generated if R V is finitely generated. In this note all vertex algebras are assumed to be finitely strongly generated. V is called lisse (or C 2 -cofinite) if dim X V = 0. By (4), it follows that V is lisse if and only if dim Spec(gr V ) = 0 ([A12a]). Hence lisse vertex algebras can be regarded as an analogue of finite-dimensional algebras. For instance, consider the case V = V k (g). We have F 1 V k (g)) = g[t 1 ]t 2 V k (g), and there is an isomorphism of Poisson algebras Hence (5) C[g ] R V, x 1... x r (x 1 t 1 )... (x r t 1 ) 0 (x i g). X V k (g) = g. Also, we have the isomorphism Spec(gr V k (g)) = J g. By (5), we have X Lk (g) g, which is G-invariant and conic. It is known [DM06] that (6) L k (g) is lisse L k (g) is integrable as a ĝ-module ( k Z 0 ). Hence the lisse condition may be regarded as a generalization of the integrability condition to an arbitrary vertex algebra. A vertex algebra is called conformal if there exists a vector ω, called the conformal vector, such that the corresponding field ω(z) = n Z L nz n 2 satisfies the following conditions. (1) [L m, L n ] = (m n)l m+n + m3 m 12 δ m+n,0 c id V, where c is a constant called the central charge of V; (2) L 0 acts semisimply on V; (3) L 1 = T. For a conformal vertex algebra V we set V = {v V L 0 v = V }, so 2 This filtration is separated if V is non-negatively graded, which we assume.

4 4 TOMOYUKI ARAKAWA that V = V. The universal affine vertex algebra V k (g) is conformal by the Sugawara construction provided that k h. A positive energy representation M of a conformal vertex algebra V is a V -module M on which L 0 acts diagonally and the L 0 -eivenvalues on M is bounded from below. An ordinary representation is a positive energy representation such that each L 0 -eivenspaces are finite-dimensional. For a finitely generated ordinary representation M, the normalized character χ V (q) = tr V (q L0 c/24 ) is well-defined. For a conformal vertex algebra V = V, one defines Zhu s algebra [FZ92] Zhu(V ) of V by Zhu(V ) = V/V V, V V = span C {a b a, b V }, where a b = ( ) a i 0 a i (i 2) b for a a. Zhu(V ) is a unital associative algebra by the multiplication a b = ( ) a i 0 a i (i 1) b. There is a bijection between the isomorphism classes Irrep(V ) of simple positive energy representation of V and that of simple Zhu(V )-modules ([FZ92, Zhu96]). The grading of V gives a filtration on Zhu(V ) which makes it quasi-commutative, and there is a surjective map (7) R V gr Zhu(V ) of Poisson algebras. Hence, if V is lisse then Zhu(V ) is finite-dimensional, so there are only finitely many irreducible positive energy representations of V. Moreover, the lisse condition implies that any simple V -module is a positive energy representation ([ABD04]). A conformal vertex algebra is called rational if any positive energy representation of V is completely reducible. For instance, the simple affine vertex algebra L k (g) is rational if and only if L k (g) is integrable, and if this is the case L k (g) -Mod is exactly the category of integrable representations of ĝ at level k. A theorem of Y. Zhu [Zhu96] states that if V is a rational, lisse, Z 0 -graded conformal vertex algebra such that V 0 = C 0, then the character χ M (e 2πiτ ) converges to a holomorphic functor on the upper half plane for any M Irrep(V ). Moreover, the space spanned by the characters χ M (e 2πiτ ), M Irrep(V ), is invariant under the natural action of SL 2 (Z). This theorem was strengthened in [DLN15] to the fact that {χ M (e 2πiτ ) M Irrep(V )} forms a vector valued modular function by showing the congruence property. Furthermore, it has been shown in [Hua08] that the category of V - modules form a modular tensor category. 3. W -algebras W -algebras are defined by the method of the quantized Drinfeld-Sokolov reduction that was discovered by Feigin and Fenkel [FF90]. In the most general definition of W -algebras given by Kac and Wakimoto [KRW03], W -algebras are associated with the pair (g, f) of a simple Lie algebra g and a nilpotent element f g. The corresponding W -algebra is a one-parameter family of vertex algebra denoted by W k (g, f), k C. By definition, W k (g, f) := H 0 DS,f (V k (g)),

5 REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 5 where H DS,f (M) denotes the BRST cohomology of the quantized Drinfeld-Sokolov reduction associated with (g, f) with coefficient in a V k (g)-module M, which is defined as follows. Let {e, h, f} be an sl 2 -triple associated with f, g j = {x g [h, x] = 2j}, so that g = j 1 2 Z g j. Set g 1 = j 1 g j, g >0 = j 1/2 g j. Then χ : g 1 [t, t 1 ] C, xt n δ n, 1 (f x), defines a character. Let F χ = U(g >0 [t, t 1 ]) U(g>0[t]+g 1 [t,t 1 ]) C χ, where C χ is the one-dimensional representation of g >0 [t] + g 1 [t, t 1 ] on which g 1 [t, t 1 ] acts by the character χ and g >0 [t] acts triviality. Then, for a V k (g)-module M, H DS,f (M) = H 2 + (g >0 [t, t 1 ], M F χ ), where H 2 + (g >0 [t, t 1 ], N) is the semi-infinite g >0 [t, t 1 ]-cohomology [Feĭ84] with coefficient in a g >0 [t, t 1 ]-module N. Since it is defined by a BRST cohomology, W k (g, f) is naturally a vertex algebra, which is called W-algebra associated with (g, f) at level k. By [FBZ04, KW04], we know that HDS,f i (V k (g)) = 0 for i 0. If f = 0 we have by definition W k (g, f) = V k (g). The W -algebra W k (g, f) is conformal provided that k h. Let S f = f +g e g = g, the Slodowy slice at f, where g e denotes the centralizer of e in g. The affine variety S f has a Poisson structure obtained from that of g by Hamiltonian reduction ([GG02]). We have (8) X Wk (g,f) = S f, ([DSK06, A15a]). Also, we have Spec(gr W k (g, f)) = J S f Zhu(W k (g, f)) = U(g, f) ([A07, DSK06]), where U(g, f) is the finite W -algebra associated with (g, f) ([Pre02]). Therefore, the W -algebra W k (g, f) can be regarded as an affinization of the finite W -algebra U(g, f). The map (7) for W k (g, f) is an isomorphism, which recovers the fact [Pre02, GG02] that U(g, f) is a quantization of the Slodowy slice S f. The definition of W k (g, f) naturally extends [KRW03] to the case that g is a (basic classical) Lie superalgebra and f is a nilpotent element in the even part of g. We have W k (g, f) = W k (g, f ) if f and f belong to the same nilpotent orbit of g. The W -algebra associated with a minimal nilpotent element f mim and a principal nilpotent element f prin are called a minimal W -algebra and a principal W -algebra, respectively. For g = sl 2, these two coincide and are isomorphic to the Virasoro vertex algebra of central charge 1 6(k 1) 2 /(k + 2) provided that k 2. In [KRW03] it was shown that almost every superconformal algebra appears as the minimal W -algebra W k (g, f min ) for some Lie superalgebra g, by describing the generators and the relations (OPEs) of minimal W -algebras. Except for some special cases, the presentation of W k (g, f) by generators and relations is not known for other nilpotent elements. Historically, the principal W -algebras were first extensively studied (see [BS95]). In the case that g = sl n, the non-critical principal W -algebras are isomorphic to the Fateev-Lukyanov s W n -algebra [FL88] ([FF90, FBZ04]). The critical principal W -algebra W h (g, f prin ) is isomorphic to the Feigin-Frenkel center z(ĝ) of ĝ, that is the center of the critical affine vertex algebra V h (g) ([FF92]). For a general f, we have the isomorphism z(ĝ) = Z(W h (g, f)),

6 6 TOMOYUKI ARAKAWA ([A12b, AMor]), where Z(W h (g, f)) denotes the center of W h (g, f). This fact has an application to Vinberg s Problem for the centralizer g e of e in g [AP15]. 4. Representation theory of W -algebras The definition of W k (g, f) by the quantized Drinfeld-Sokolov reduction gives rise to a functor V k (g) -Mod W k (g, f) -Mod M H 0 DS,f (M). Let O k be the category O of ĝ at level k. Then O k is naturally considered as a full subcategory of V k (g) -Mod. For a weight λ of ĝ of level k, let L(λ) be the irreducible highest weight representations of ĝ with highest weight λ. Theorem 1 ([A05]). Let f min O min and let k be an arbitrary complex number. (i) We have H i DS,f min (M) = 0 for any M O k and i Z\{0}. Therefore, the functor O k W k (g, f min ) -Mod, M H 0 DS,f min (M) is exact. (ii) For a weight λ of ĝ of level k, H 0 DS,f min (L(λ)) is zero or isomorphic to an irreducible highest weight representation of W k (g, f min ). Moreover, any irreducible highest weight representation the minimal W -algebra W k (g, f min ) arises in this way. By Theorem 1 and the Euler-Poincaré principal, the character ch H 0 DS,f θ (L(λ)) is expressed in terms of ch L(λ). Since ch L(λ) is known [KT00] for all non-critical weight λ, Theorem 1 determines the character of all non-critical irreducible highest weight representation of W k (g, f min ). In the case that k is critical the character of irreducible highest weight representation of W k (g, f min ) is determined by the Lusztig-Feigin-Frenkel conjecture ([Lus91, AF12, FG09]). Remark 2. Theorem 1 holds in the case that g is a basic classical Lie superalgebra as well. In particular one obtains the character of irreducible highest weight representations of superconformal algebras that appear as W k (g, f min ) once the character of irreducible highest weight representations of ĝ is given. Let KL k be the full subcategory of O k consisting of objects on which g[t] acts locally finitely. Altough the functor (9) O k W k (g, f) -Mod, M H 0 DS,f (M) is not exact for a general nilpotent element f, we have the following result. Theorem 3 ([A15a]). Let f, k be arbitrary. We have H i DS,f min (M) = 0 for any M KL k and i 0. Therefore, the functor is exact. KL k W k (g, f min ) -Mod, M H 0 DS,f min (M) In the case that f is a principal nilpotent element, Theorem 3 has been proved in [FG10] using Theorem 4 below.

7 REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 7 The restriction of the quantized Drinfeld-Sokolv reduction functor to KL k does not produces all the irreducible highest weight representations of W k (g, f). However, one can modify the functor (9) to the -reduction functor H,f 0 (?) (defined in [FKW92]) to obtain the following result for the principal W -algebras. Theorem 4 ([A07]). Let f be a principal nilpotent element, and let k be an arbitrary complex number. (i) We have H i,f prin (M) = 0 for any M O k and i Z\{0}. Therefore, the functor O k W k (g, f prin ) -Mod, M H i,f prin (M) is exact. (ii) For a weight λ of ĝ of level k, H 0,f prin (L(λ)) is zero or isomorphic to an irreducible highest weight representation of W k (g, f prin ). Moreover, any irreducible highest weight representation the principal W -algebra W k (g, f prin ) arises in this way. In type A we can derive the similar result as Theorem 4 for any nilpotent element f using the work of Brundan and Kleshchev [BK08] on the representation theory of finite W-algebras ([A11]). In particular the character of all ordinary irreducible representations of W k (sl n, f) has been determined for a non-critical k. 5. BRST reduction of associated varieties Let W k (g, f) be the unique simple graded quotient of W k (g, f). The associated variety X Wk (g,f) is a subvariety of X Wk (g,f) = S f, which is invariant under the natural C -action on S f that contracts to the point f S f. Therefore W k (g, f) is lisse if and only if X Wk (g,f) = {f}. By Theorem 3, W k (g, f) is a quotient of the vertex algebra H 0 DS,f (L k(g)), provided that it is nonzero. Theorem 5 ([A15a]). For any f g and k C we have Therefore, X H 0 DS,f (L k (g)) = X Lk (g) S f. (i) H 0 DS,f (L k(g)) 0 if and only if G.f X Lk (g); (ii) If X Lk (g) = G.f then X H 0 DS,f (L k (g)) = {f}. Hence H 0 DS,f (L k(g)) is lisse, and thus, so is its quotient W k (g, f). Theorem 5 can be regarded as a vertex algebra analogue of the corresponding result [Los11, Gin08] for finite W -algebras. Note that if L k (g) is integrable we have HDS,f 0 (L k(g)) = 0 by (6). Therefore we need to study more general representations of ĝ to obtain lisse W -algebras using Theorem 5. Recall that the irreducible highest weight representation L(λ) of ĝ is called admissible [KW89] (1) if λ is regular dominant, that is, λ + ρ, α Z 0 for any α re +, and (2) Q (λ) = Q re. Here re is the set of real roots of ĝ, re + the set of positive real roots of ĝ, and (λ) = {α re λ + ρ, α Z}, the set of integral roots if λ. Admissible representations are (conjecturally all) modular invariant representations of ĝ, that is, the characters of admissible representations are

8 8 TOMOYUKI ARAKAWA invariant under the natural action of SL 2 (Z) ([KW88]). The simple affine vertex algebra L k (g) is admissible as a ĝ-module if and only if { k + h = p q, p, q N, (p, q) = 1, p h if (q, r ) = 1, (10) h if (q, r ) = r ([KW08]). Here h is the Coxeter number of g and r is the lacity of g. If this is the case k is called an admissible number for ĝ and L k (g) is called an admissible affine vertex algebra. Theorem 6 ([A15a]). Let L k (g) be an admissible affine vertex algebra. (i) (Feigin-Frenkel conjectrue) We have X Lk (g) N. (ii) The variety X Lk (g) is irreducible. That is, there exists a nilpotent orbit O k of g such that X Lk (g) = O k. (iii) More precisely, let k be an admissible number of the form (10). Then { {x g (ad x) 2q = 0} if (q, r ) = 1, X Lk (g) = {x g π θs (x) 2q/r = 0} if (q, r ) = r, where θ s is the highest short root of g and π θs is the irreducible finitedimensional representation of g with highest weight θ s. From Theorem 5 and Theorem 6 we immediately obtain the following assertion, which was (essentially) conjectured by Kac and Wakimoto [KW08]. Theorem 7 ([A15a]). Let L k (g) be an admissible affine vertex algebra, and let f O k. Then the simple affine W -algebra W k (g, f) is lisse. In the case that X Lk (g) = G.f prin, the lisse W -algebras obtained in Theorem 7 is the minimal series principal W -algebras studied in [FKW92]. In the case that g = sl 2, these are exactly the minimal series Virasoro vertex algebras ([FF84, BFM, Wan93]). The Frenkel-Kac-Wakimoto conjecture states that these minimal series principal W -algebras are rational. More generally, all the lisse W -algebras W k (g, f) that appear in Theorem 7 are conjectured to be rational ([KW08, A15a]). 6. The rationality of minimal series principal W-algerbas An admissible affine vertex algebra L k (g) is called non-degenearte ([FKW92]) if X Lk (g) = N = G.f prin. If this is the case k is called a non-degenerate admissible number for ĝ. By theorem 6 (iii), most admissible affine vertex algebras are non-degenerate. More precisely, an { admissible number k of the form (10) is non-degenerate if and only if q h if (q, r ) = 1, r L h if (q, r ) = r, where L h is the dual Coxeter number of the Langlands dual Lie algebra L g. For a non-degenerate admissible number k, the simple principal W -algebra W k (g, f prin ) is lisse by Theorem 7. The following assertion settles the Frenkel-Kac-Waimoto conjecture [FKW92] in full generality. Theorem 8 ([A15b]). Let k be a non-degenerate admissible number. simple principal W -algebra W k (g, f prin ) is rational. Then the

9 REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 9 The proof of Theorem 8 based on Theorem 4, Theorem 7, and the following assertion on admissible affine vertex algebras, which was conjectured by Adamović and Milas [AM95]. Theorem 9 ([A16]). Let L k (g) be an admissible affine vertex algebra. Then L k (g) is rational in the category O, that is, any L k (g)-module that belongs to O is completely reducible. The following assertion, that has been widely believed since [KW90], gives a yet another realization of minimal series principal W -algebras. Theorem 10 ([ACL]). Let g be simply laced. For an admissible affine vertex algebra L k (g), the vertex algebra (L k (g) L 1 (g)) g[t] is isomorphic to a minimal series principal W -algebra. Conversely, any minimal series principal W -algebra associated with g appears in this way. In the case that g = sl 2 and k is a non-negative integer, the statement of Theorem 10 is well-known as the GKO construction of the discrete series of the Virasoro vertex algebras [GKO86]. Some partial results have been obtained previously in [ALY17, AJ17]. From Theorem 10, it follows that the minimal series principal W -algebra W p/q h (g, f prin ) of ADE type is unitary, that is, any simple W p/q h (g, f prin )-module is unitary in the sense of [DLN14], if and only if p q = Four-dimensional N = 2 superconformal algebras, Higgs branch conjecture and the class S chiral algebras In the study of four-dimensional N = 2 superconformal field theories in physics, Beem, Lemos, Liendo, Peelaers, Rastelli, and van Rees [BLL + 15] have constructed a remarkable map (11) Φ : {4d N = 2 SCFTs} {vertex algebras} such that, among other things, the character of the vertex algebra Φ(T ) coincides with the Schur index of the corresponding 4d N = 2 SCFT T, which is an important invariant of the theory T. How do vertex algebras coming from 4d N = 2 SCFTs look like? According to [BLL + 15], we have c 2d = 12c 4d, where c 4d and c 2d are central charges of the 4d N = 2 SCFT and the corresponding vertex algebra, respectively. Since the central charge is positive for a unitary theory, this implies that the vertex algebras obtained by Φ are never unitary. In particular integrable affine vertex algebras never appear by this correspondence. The main examples of vertex algebras considered in [BLL + 15] are the affine vertex algebras L k (g) of types D 4, F 4, E 6, E 7, E 8 at level k = h /6 1, which are non-rational, non-admissible affine vertex algebras studies in [AMor16]. One can find more examples in the literature, see e.g. [BPRvR15, XYY16, CS16, SXY17]. Now, there is another important invariant of a 4d N = 2 SCFT T, called the Higgs branch, which we denote by Higgs T. The Higgs branch Higgs T is an affine algebraic variety that has the hyperkähler structure in its smooth part. In particular, Higgs T is a (possibly singular) symplectic variety. Let T be one of the 4d N = 2 SCFTs studied in [BLL + 15] such that that Φ(T ) = L k (g) with k = h /6 1 for types D 4, F 4, E 6, E 7, E 8 as above. It

10 10 TOMOYUKI ARAKAWA is known that Higgs T = O min, and this equals [AMor16] to the the associated variety X Φ(T ). It is expected that this is not just a coincidence. Conjecture 11 (Beem and Rastelli [BR17]). For a 4d N = 2 SCFT T, we have Higgs T = X Φ(T ). So we are expected to recover the Higgs branch of a 4d N = 2 SCFT from the corresponding vertex algebra, which is a purely algebraic object. We note that Conjecture 11 is a physical conjecture since the Higgs branch is not a mathematical defined object at the moment. The Schur index is not a mathematical defined object either. However, in view of (11) and Conjecture 11, one can try to define both Higgs branches and Schur indices of 4d N = 2 SCFTs using vertex algebras. We note that there is a close relationship between Higgs branches of 4d N = 2 SCFTs and Coulomb branches of three-dimensional N = 4 gauge theories whose mathematical definition has been given by Braverman, Finkelberg and Nakajima [BFN16b]. Although Higgs branches are symplectic varieties, the associated variety X V of a vertex algebra V is only a Poisson variety in general. A vertex algebra V is called quasi-lisse ([AK17]) if X V has only finitely many symplectic leaves. If this is the case symplectic leaves in X V are algebraic ([BG03]). Clearly, lisse vertex algebras are quasi-lisse. The simple affine vertex algebra L k (g) is quasi-lisse if and only if X Lk (g) N. In particular, admissible affine vertex algebras are quasi-lisse. See [AMor16, AMor17a, AMor17b] for more examples of quasi-lisse vertex algebras. Physical intuition expects that vertex algebras that come from 4d N = 2 SCFTs via the map Φ are quasi-lisse. By extending Zhu s argument [Zhu96] using a theorem of Etingof and Schelder [ES10], we obtain the following assertion. Theorem 12 ([AK17]). Let V be a quasi-lisse Z 0 -graded conformal vertex algebra such that V 0 = C. Then there only finitely many simple ordinary V -modules. Moreover, for an ordinary V -module M, the character χ M (q) satisfies a modular linear differential equation. Since the space of solutions of a modular linear differential equation is invariant under the action of SL 2 (Z), Theorem 12 implies that a quasi-lisse vertex algebra possesses a certain modular invariance property, although we do not claim that the normalized characters of ordinary V -modules span the space of the solutions. Note that Theorem 12 implies that the Schur indices of 4d N = 2 SCFTs have some modular invariance property. This is something that has been conjectured by physicists ([BR17]). There is a distinct class of four-dimensional N = 2 superconformal field theories called the theory of class S ([Gai12, GMN13]), where S stands for 6. The vertex algebras obtained from the theory of class S is called the chiral algebras of class S ([BPRvR15]). The Moore-Tachikawa conjecture [MT12], which was recently proved in [BFN17], describes the Higgs branches of the theory of class S in terms of twodimensional topological quantum field theories. Let V be the category of vertex algebras, whose objects are semisimple groups, and Hom(G 1, G 2 ) is the isomorphism classes of conformal vertex algebras V with a vertex algebra homomorphism V h 1 (g1 ) V h 2 (g2 ) V

11 REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 11 such that the action of g 1 [t] g 2 [t] on V is locally finite. Here g i = Lie(G i ) and h i is the dual Coxeter number of g i in the case that g i is simple. If g i is not simple we understand V h i (gi ) to be the tensor product of the critical level universal affine vertex algebras corresponding to all simple components of g i. The composition V 1 V 2 of V 1 Hom(G 1, G 2 ) and V 2 Hom(G 1, G 2 ) is given by the relative semiinfinite cohomology V 1 V 2 = H 2 +0 (ĝ 2, g 2, V 1 V 2 ), where ĝ 2 denotes the direct sum of the affine Kac-Moody algebra associated with the simple components of g 2. By a result of [AG02], one finds that the identity morphism id G is the algebra DG ch of chiral differential operators on G ([MSV99, BD04]) at the critical level, whose associated variety is canonically isomorphic to T G. The following theorem, which was conjectured in [BLL + 15] (see [Tac1, Tac2]) for mathematical expositions), describes the chiral algebras of class S. Theorem 13 ([A17]). Let B 2 the category of 2-bordisms. There exists a unique monoidal functor η G : B 2 V which sends (1) the object S 1 to G, (2) the cylinder, which is the identity morphism id S 1, to the identity morphism id G = DG ch, and (3) the cap to HDS,f 0 prin (DG ch). Moreover, we have X η G (B) = ηg BF N (B) for any 2-bordism B, where ηg BF N is the functor form B 2 to the category of symplectic variaties constructed in [BFN17]. The last assertion of the above theorem confirms the Higgs branch conjecture for the theory of class S. References [ABD04] Toshiyuki Abe, Geoffrey Buhl, and Chongying Dong. Rationality, regularity, and C 2 -cofiniteness. Trans. Amer. Math. Soc., 356(8): (electronic), [AM95] Dražen Adamović and Antun Milas. Vertex operator algebras associated to modular invariant representations for A (1) 1. Math. Res. Lett., 2(5): , [AGT10] Luis F. Alday, Davide Gaiotto, and Yuji Tachikawa. Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys., 91(2): , [AFO17] Mina Aganagic, Edward Frenkel, and Andrei Okounkov. Quantum q-langlands correspondence. arxiv: [hep-th]. [A05] Tomoyuki Arakawa. Representation theory of superconformal algebras and the Kac- Roan-Wakimoto conjecture. Duke Math. J., 130(3): , [A07] Tomoyuki Arakawa. Representation theory of W -algebras. Invent. Math., 169(2): , [A11] Tomoyuki Arakawa. Representation theory of W -algebras, II. In Exploring new structures and natural constructions in mathematical physics, volume 61 of Adv. Stud. Pure Math., pages Math. Soc. Japan, Tokyo, [A12a] Tomoyuki Arakawa. A remark on the C 2 cofiniteness condition on vertex algebras. Math. Z., 270(1-2): , [A12b] Tomoyuki Arakawa. W-algebras at the critical level. Contemp. Math., 565:1 14, [A15a] Tomoyuki Arakawa. Associated varieties of modules over Kac-Moody algebras and C 2 -cofiniteness of W-algebras. Int. Math. Res. Not., 2015: , [A15b] Tomoyuki Arakawa. Rationality of W-algebras: principal nilpotent cases. Ann. Math., 182(2): , [A16] Tomoyuki Arakawa. Rationality of admissible affine vertex algebras in the category O. Duke Math. J., 165(1):67 93, [A17] Tomoyuki Arakawa. Chiral algebras of class S and symplectic varieties. in preparation. [ACL] Tomoyuki Arakawa, Thomas Creutzig, and Andrew R. Linshaw. W-algebras as coset vertex algebras. in preparation.

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