Representation Theory of W-Algebras and Higgs Branch Conjecture

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 infinite- dimensional 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 Christo- pher Beem, Davide Gaiotto, Madalena Lemos, Victor Kac, Anne Moreau, Hi- raku 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, 17K18724.

2. 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 ⊗ 7→ −n−1 a b a(z)b = ( n∈Z a(n)z )b, such that the following are satisﬁed:

• (Unit axioms) (|0⟩)(z) = 1V 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 TOMOYUKI ARAKAWA

The operator T : a 7→ a(−2)|0⟩ is called the translation operator and it satisfies (T a)(z) = [T, a(z)] = ∂za(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: ∑ ( ) m (1) [a , b ] = (a b) − , (m) (n) j (j) (m+n j) j≥0 ∑ ( ) j m m (2) (a b) = (−1) (a − b − (−1) b − a ). (m) (n) j (m j) (n+j) (m+n j) (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 bg = g[t, t−1]⊕ CK be the affine Kac-Moody algebra associated with g. The commutation relations of bg are given by m n m+n (3) [xt , yt ] = [x, y]t + mδm+n,0(x|y)K, [K, bg] = 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 k b V (g) = U(g) ⊗U(g[t]⊕CK) Ck, where Ck 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) = (xtn)z−n−1 (x ∈ g). n∈Z Here on the left-hand-side g is considered as a subspace of V k(g) by the embedding g ,→ V k(g), x 7→ (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 bg-module of level k. Conversely, any smooth bg-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 bg-modules of level k. Let Lk(g) be the unique simple graded quotient of V k(g), which is isomorphic to the irreducible highest weight representation L(kΛ0) with highest weight kΛ0 as a bg-module. The vertex algebra Lk(g) is called the simple affine vertex algebra associated with g at level k k, and Lk(g) -Mod forms a full subcategory of V (g) -Mod, the category of smooth bg-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 same1 as differential algebras ([Bor86]).

1However, the modules of a commutative vertex algebra are not the same as the modules as a diﬀerential algebra. 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, 2 and hence can be regarded⊕ as a quantization of the associated graded Poisson p p+1 • vertex algebra gr V = p F V/F V , where F V is the canonical filtration of V . By definition, ∑ p { | ⟩ | ∈ ≥ ≥ } F V = spanC (a1)(−n1−1) ... (ar)(−nr −1) 0 ai V, ni 0, ni p . i The subspace 1 0 1 RV := V/F V = F V/F V ⊂ gr V is called Zhu’s C2-algebra of V . The Poisson vertex algebra structure of gr V re- stricts to the Poisson algebra structure of RV , which is given by ¯ ¯ a.¯ b = a(−1)b, {a,¯ b} = a(0)b. The Poisson variety

XV = Specm(RV ) called the associated variety of V ([A12a]). We have [Li05] the inclusion

(4) Specm(gr V ) ⊂ J∞XV .

A vertex algebra V is called finitely strongly generated if RV is finitely generated. In this note all vertex algebras are assumed to be finitely strongly generated. V is called lisse (or C2-cofinite) if dim XV = 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 1V k(g)) = g[t−1]t−2V k(g), and there is an isomorphism of Poisson algebras ∗ ∼ −1 −1 C[g ] → RV , x1 . . . xr 7→ (x1t ) ... (xrt )|0⟩ (xi ∈ g). Hence ∼ ∗ (5) XV k(g) = g . k ∼ ∗ ⊂ Also, we have the isomorphism Spec(gr V (g)) = J∞g . By (5), we have XLk(g) g∗, which is G-invariant and conic. It is known [DM06] that

(6) Lk(g) is lisse ⇐⇒ Lk(g) is integrable as a bg-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 con- −n−2 formal vector, such that the corresponding ﬁeld ω(z) = n∈Z Lnz satisﬁes − m3−m the following conditions. (1) [Lm,Ln] = (m n)Lm+n + 12 δm+n,0c idV , where c is a constant called the central charge of V; (2) L0 acts semisimply on V; (3) L−1 = T . For a conformal vertex algebra V we set V∆ = {v ∈ V | L0v = ∆V }, so

2This filtration is separated if V is non-negatively graded, which we assume. 4 TOMOYUKI ARAKAWA ⊕ k that V = ∆ V∆. The universal affine vertex algebra V (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 L0 acts diagonally and the L0-eivenvalues on M is bounded from below. An ordinary representation is a positive energy representation such that each L0-eivenspaces are finite-dimensional. For a finitely generated ordinary representation M, the normalized character

L0−c/24 χV (q) = trV (q ) 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 = spanC{a ◦ b | a, b ∈ V }, ( ) ∑ ◦ ∆a ∈ where a b = i≥0 a(i−2)b for a ∆a. Zhu(V ) is a unital associative algebra i ( ) ∑ ∆a by the multiplication a ∗ b = a − b. There is a bijection between the i≥0 i (i 1) 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) RV ↠ 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 Lk(g) is rational if and only if Lk(g) is integrable, and if this is the case Lk(g) -Mod is exactly the category of integrable representations of bg at level k. A theorem of Y. Zhu [Zhu96] states that if V is a rational, lisse, Z≥0-graded conformal vertex algebra 2πiτ such that V0 = C|0⟩, then the character χM (e ) converges to a holomorphic functor on the upper half plane for any M ∈ Irrep(V ). Moreover, the space spanned 2πiτ by the characters χM (e ), M ∈ Irrep(V ), is invariant under the natural action of 2πiτ SL2(Z). This theorem was strengthened in [DLN15] to the fact that {χM (e ) | 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 Wk(g, f), k ∈ C. By definition, Wk 0 k (g, f) := HDS,f (V (g)), REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 5

• where HDS,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 sl2-triple associated⊕ with f, g⊕j = x g | [h, x] = 2j}, so that g = ∈ 1 Z gj. Set g≥1 = ≥ gj, g>0 = ≥ gj. j 2 j 1 j 1/2 −1 n Then χ : g≥1[t, t ] → C, xt 7→ δn,−1(f|x), defines a character. Let Fχ = −1 ⊗ −1 C C U(g>0[t, t ]) U(g>0[t]+g≥1[t,t ]) χ, where χ is the one-dimensional represen- −1 −1 tation of g>0[t] + g≥1[t, t ] on which g≥1[t, t ] acts by the character χ and g>0[t] acts triviality. Then, for a V k(g)-module M, • ∞ • − 2 + 1 ⊗ HDS,f (M) = H (g>0[t, t ],M Fχ), ∞ +• −1 −1 where H 2 (g>0[t, t ],N) is the semi-infinite g>0[t, t ]-cohomology [Fe˘ı84]with −1 coefficient in a g>0[t, t ]-module N. Since it is defined by a BRST cohomology, Wk(g, f) is naturally a vertex algebra, which is called W-algebra associated with i k ̸ (g, f) at level k. By [FBZ04, KW04], we know that HDS,f (V (g)) = 0 for i = 0. If f = 0 we have by definition Wk(g, f) = V k(g). The W -algebra Wk(g, f) is conformal provided that k ≠ −h∨. e ∼ ∗ e Let Sf = f +g ⊂ g = g , the Slodowy slice at f, where g denotes the centralizer ∗ of e in g. The affine variety Sf has a Poisson structure obtained from that of g by Hamiltonian reduction ([GG02]). We have ∼ S Wk ∼ S (8) XWk(g,f) = f , Spec(gr (g, f)) = J∞ f ([DSK06, A15a]). Also, we have k ∼ Zhu(W (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 Wk(g, f) can be regarded as an affinization of the finite W -algebra U(g, f). The map (7) for Wk(g, f) is an isomorphism, which recovers the fact [Pre02, GG02] that U(g, f) is a quantization of the Slodowy slice Sf . The definition of Wk(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 Wk(g, f) = Wk(g, f ) if f and f belong to the same nilpotent orbit of g. The W -algebra associated with a minimal nilpotent element fmim and a principal nilpotent element fprin are called a minimal W -algebra and a principal W -algebra, respectively. For g = sl2, 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 k appears as the minimal W -algebra W (g, fmin) 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 Wk(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 = sln, the non-critical principal W -algebras are isomorphic to the Fateev-Lukyanov’s Wn-algebra [FL88] ([FF90, FBZ04]). The critical principal −h∨ W -algebra W (g, fprin) is isomorphic to the Feigin-Frenkel center z(bg) of bg, that ∨ is the center of the critical affine vertex algebra V −h (g) ([FF92]). For a general f, we have the isomorphism ∼ −h∨ z(bg) = Z(W (g, f)), 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 ge of e in g [AP15].

4. Representation theory of W -algebras The deﬁnition of Wk(g, f) by the quantized Drinfeld-Sokolov reduction gives rise to a functor V k(g) -Mod → Wk(g, f) -Mod 7→ 0 M HDS,f (M).

Let Ok be the category O of bg at level k. Then Ok is naturally considered as a full subcategory of V k(g) -Mod. For a weight λ of bg of level k, let L(λ) be the irreducible highest weight representations of bg with highest weight λ.

Theorem 1 ([A05]). Let fmin ∈ Omin and let k be an arbitrary complex number. (i) We have Hi (M) = 0 for any M ∈ O and i ∈ Z\{0}. Therefore, DS,fmin k the functor O → Wk( , f ) -Mod,M 7→ H0 (M) k g min DS,fmin is exact. (ii) For a weight λ of bg of level k, H0 (L(λ)) is zero or isomorphic to an DS,fmin k irreducible highest weight representation of W (g, fmin). Moreover, any ir- k reducible highest weight representation the minimal W -algebra W (g, fmin) arises in this way. By Theorem 1 and the Euler-Poincar´eprincipal, the character ch H0 (L(λ)) DS,fθ 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 k weight representation of W (g, fmin). In the case that k is critical the character k of irreducible highest weight representation of W (g, fmin) 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 k representations of superconformal algebras that appear as W (g, fmin) once the character of irreducible highest weight representations of bg is given.

Let KLk be the full subcategory of Ok consisting of objects on which g[t] acts locally ﬁnitely. Altough the functor O → Wk 7→ 0 (9) k (g, f) -Mod,M HDS,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 Hi (M) = 0 for any DS,fmin M ∈ KLk and i ≠ 0. Therefore, the functor KL → Wk( , f ) -Mod,M 7→ H0 (M) k g min DS,fmin is exact. In the case that f is a principal nilpotent element, Theorem 3 has been proved in [FG10] using Theorem 4 below. REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 7

The restriction of the quantized Drinfeld-Sokolv reduction functor to KLk does not produces all the irreducible highest weight representations of Wk(g, f). How- − 0 ever, one can modify the functor (9) to the “ ”-reduction functor H−,f (?) (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 Hi (M) = 0 for any M ∈ O and i ∈ Z\{0}. Therefore, the −,fprin k functor O → Wk( , f ) -Mod,M 7→ Hi (M) k g prin −,fprin is exact. (ii) For a weight λ of bg of level k, H0 (L(λ)) is zero or isomorphic to an −,fprin k irreducible highest weight representation of W (g, fprin). Moreover, any ir- k reducible highest weight representation the principal W -algebra W (g, fprin) 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 k representations of W (sln, f) has been determined for a non-critical k.

5. BRST reduction of associated varieties k Let Wk(g, f) be the unique simple graded quotient of W (g, f). The associated

k S variety XWk(g,f) is a subvariety of XW (g,f) = f , which is invariant under the ∗ natural C -action on Sf that contracts to the point f ∈ Sf . Therefore Wk(g, f) is { } lisse if and only if XWk(g,f) = f . W 0 By Theorem 3, k(g, f) is a quotient of the vertex algebra HDS,f (Lk(g)), provided that it is nonzero. Theorem 5 ([A15a]). For any f ∈ g and k ∈ C we have ∼ X 0 = X ∩ Sf . HDS,f (Lk(g)) Lk(g) Therefore, 0 ̸ ⊂ (i) HDS,f (Lk(g)) = 0 if and only if G.f XLk(g); 0 (ii) If X = G.f then X 0 = {f}. Hence H (Lk(g)) is lisse, Lk(g) HDS,f (Lk(g)) DS,f and thus, so is its quotient Wk(g, f). Theorem 5 can be regarded as a vertex algebra analogue of the corresponding result [Los11, Gin08] for finite W -algebras. 0 Note that if Lk(g) is integrable we have HDS,f (Lk(g)) = 0 by (6). Therefore we need to study more general representations of bg to obtain lisse W -algebras using Theorem 5. Recall that the irreducible highest weight representation L(λ) of bg is called ad- ∨ missible [KW89] (1) if λ is regular dominant, that is, ⟨λ + ρ, α ⟩ ̸∈ −Z≥0 for any ∈ re Q Q re re b re α ∆+ , and (2) ∆(λ) = ∆ . Here ∆ is the set of real roots of g, ∆+ the set of positive real roots of bg, and ∆(λ) = {α ∈ ∆re | ⟨λ + ρ, α∨⟩ ∈ Z}, the set of integral roots if λ. Admissible representations are (conjecturally all) modular invariant representations of bg, that is, the characters of admissible representations are 8 TOMOYUKI ARAKAWA invariant under the natural action of SL2(Z) ([KW88]). The simple affine vertex algebra L (g) is admissible as a bg-module if and only if k { p h∨ if (q, r∨) = 1, (10) k + h∨ = , p, q ∈ N, (p, q) = 1, p ≥ q 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 bg and Lk(g) is called an admissible affine vertex algebra.

Theorem 6 ([A15a]). Let Lk(g) be an admissible aﬃne vertex algebra. ⊂ N (i) (Feigin-Frenkel conjectrue) We have XLk(g) . O (ii) The variety XLk(g) is irreducible. That is, there exists a nilpotent orbit k of g such that O XLk(g) = 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) { ∈ | 2q/r } ∨ ∨ x g πθs (x) = 0 if (q, r ) = r ,

where θs is the highest short root of g and πθs is the irreducible ﬁnite- dimensional representation of g with highest weight θs. 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 Lk(g) be an admissible aﬃne vertex algebra, and let f ∈ Ok. Then the simple aﬃne W -algebra Wk(g, f) is lisse.

In the case that XLk(g) = G.fprin, the lisse W -algebras obtained in Theorem 7 is the minimal series principal W -algebras studied in [FKW92]. In the case that g = sl2, 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 Wk(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 Lk(g) is called non-degenearte ([FKW92]) if N XLk(g) = = G.fprin. If this is the case k is called a non-degenerate admissible number for bg. 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, , where Lh∨ is the dual Coxeter number of the Langlands r∨Lh∨ if (q, r∨) = r∨ dual Lie algebra Lg. For a non-degenerate admissible number k, the simple principal W -algebra Wk(g, fprin) 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. Then the simple principal W -algebra Wk(g, fprin) is rational. 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 Adamovi´c and Milas [AM95].

Theorem 9 ([A16]). Let Lk(g) be an admissible aﬃne vertex algebra. Then Lk(g) is rational in the category O, that is, any Lk(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 g[t] algebra Lk(g), the vertex algebra (Lk(g) ⊗ L1(g)) 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 = sl2 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 pre- viously in [ALY17, AJ17]. From Theorem 10, it follows that the minimal series principal W -algebra Wp/q−h∨ (g, fprin) of ADE type is unitary, that is, any simple Wp/q−h∨ (g, fprin)-module is unitary in the sense of [DLN14], if and only if |p − q| = 1.

7. 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 c2d = −12c4d, where c4d and c2d 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 Lk(g) of types D4, F4, E6, E7, E8 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 HiggsT . The Higgs branch HiggsT is an affine algebraic variety that has the hyperKählerstructure in its smooth part. In particular, HiggsT 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 ) = Lk(g) with k = h /6 − 1 for types D4, F4, E6, E7, E8 as above. It 10 TOMOYUKI ARAKAWA is known that HiggsT = Omin, and this equals [AMor16] to the the associated variety XΦ(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

HiggsT = XΦ(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 XV of a vertex algebra V is only a Poisson variety in general. A vertex algebra V is called quasi-lisse ([AK17]) if XV has only finitely many symplectic leaves. If this is the case symplectic leaves in XV are algebraic ([BG03]). Clearly, lisse vertex algebras are quasi-lisse. The simple affine vertex algebra Lk(g) is quasi-lisse if and only if ⊂ N XLk(g) . 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 V0 = C. Then there only finitely many simple ordinary V -modules. Moreover, for an ordinary V -module M, the character χM (q) satisfies a modular linear differential equation. Since the space of solutions of a modular linear differential equation is invariant under the action of SL2(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 two- dimensional topological quantum field theories. Let V be the category of vertex algebras, whose objects are semisimple groups, and Hom(G1,G2) is the isomorphism classes of conformal vertex algebras V with a vertex algebra homomorphism −h∨ −h∨ V 1 (g1) ⊗ V 2 (g2) → V REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 11

⊕ ∨ such that the action of g1[t] g2[t] on V is locally finite. Here gi = Lie(Gi) and hi is the dual Coxeter number of gi in the case that gi is simple. If gi is not simple we −h∨ understand V i (gi) to be the tensor product of the critical level universal affine vertex algebras corresponding to all simple components of gi. The composition V1 ◦ V2 of V1 ∈ Hom(G1,G2) and V2 ∈ Hom(G1,G2) is given by the relative semi- infinite cohomology ∞ +0 V1 ◦ V2 = H 2 (bg2, g2,V1 ⊗ V2), where bg2 denotes the direct sum of the affine Kac-Moody algebra associated with the simple components of g2. By a result of [AG02], one finds that the identity Dch morphism idG is the algebra G 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 B2 the category of 2-bordisms. There exists a unique 1 monoidal functor ηG : B2 → V which sends (1) the object S to G, (2) the cylinder, Dch which is the identity morphism idS1 , to the identity morphism idG = G , and (3) the cap to H0 (Dch). Moreover, we have X ∼ ηBFN (B) for any DS,fprin G ηG(B) = G BFN B 2-bordism B, where ηG is the functor form 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 C2-cofiniteness. Trans. Amer. Math. Soc., 356(8):3391–3402 (electronic), 2004. [AM95] DraˇzenAdamovićand Antun Milas. Vertex operator algebras associated to modular (1) invariant representations for A1 . Math. Res. Lett., 2(5):563–575, 1995. [AGT10] Luis F. Alday, Davide Gaiotto, and Yuji Tachikawa. Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys., 91(2):167–197, 2010. [AFO17] Mina Aganagic, Edward Frenkel, and Andrei Okounkov. Quantum q-langlands correspondence. arXiv:1701.03146 [hep-th]. [A05] Tomoyuki Arakawa. Representation theory of superconformal algebras and the Kac- Roan-Wakimoto conjecture. Duke Math. J., 130(3):435–478, 2005. [A07] Tomoyuki Arakawa. Representation theory of W -algebras. Invent. Math., 169(2):219– 320, 2007. [A11] Tomoyuki Arakawa. Representation theory of W -algebras, II. In Exploring new struc- tures and natural constructions in mathematical physics, volume 61 of Adv. Stud. Pure Math., pages 51–90. Math. Soc. Japan, Tokyo, 2011. [A12a] Tomoyuki Arakawa. A remark on the C2 cofiniteness condition on vertex algebras. Math. Z., 270(1-2):559–575, 2012. [A12b] Tomoyuki Arakawa. W-algebras at the critical level. Contemp. Math., 565:1–14, 2012. [A15a] Tomoyuki Arakawa. Associated varieties of modules over Kac-Moody algebras and C2-cofiniteness of W-algebras. Int. Math. Res. Not., 2015:11605–11666, 2015. [A15b] Tomoyuki Arakawa. Rationality of W-algebras: principal nilpotent cases. Ann. Math., 182(2):565–694, 2015. [A16] Tomoyuki Arakawa. Rationality of admissible affine vertex algebras in the category O. Duke Math. J., 165(1):67–93, 2016. [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. 12 TOMOYUKI ARAKAWA

[AF12] Tomoyuki Arakawa and Peter Fiebig. On the restricted Verma modules at the critical level. Trans. Amer. Math. Soc., 364(9):4683–4712, 2012. [AJ17] Tomoyuki Arakawa and Cuipo Jiang. Coset vertex operator algebras and w-algebra. to appear in Special issue on Representation Theory in Sci. Schina. Math. [AK17] Tomoyuki Arakawa and Kazuya Kawasetsu. Quasi-lisse vertex algebras and modular linear differential equations, to appear in Kostant Memorial Volume. Birkhauser. [ALY17] Tomoyuki Arakawa, Ching Hung Lam, and Hiromichi Yamada. Parafermion vertex operator algebras and W-algebras. arXiv:1701.06229 [math.RT]. [AMol17] Tomoyuki Arakawa and Alexander Molev. Explicit generators in rectangular affine W-algebras of type A. Lett. Math. Phys., 107(1):47–59, 2017. [AMor16] Tomoyuki Arakawa and Anne Moreau. Joseph ideals and lisse minimal W-algebras. J. Inst. Math. Jussieu, published online., 2016. [AMor17a] Tomoyuki Arakawa and Anne Moreau. Sheets and associated varieties of affine vertex algebras, Adv. Math., Vol. 320, 157–209. [AMor17b] Tomoyuki Arakawa and Anne Moreau. On the irreducibility of associated varieties of W-algebras. in the special issue of J. Algebra in Honor of Efim Zelmanov on occasion of his 60th anniversary, published online. [AMor] Tomoyuki Arakawa and Anne Moreau. Arc spaces and chiral symplectic cores. in prepatation. [AP15] Tomoyuki Arakawa and Alexander Premet. Quantizing Mischchenko-Fomenko algebras via affine W-algebras. to appear in the special issue of Proceedings of Moscow Math. Society dedicated to Vinberg’s 80th birthday., 2015. [AG02] Sergey Arkhipov and Dennis Gaitsgory. Differential operators on the loop group via chiral algebras. Int. Math. Res. Not., (4):165–210, 2002. [BM13] Bojko Bakalov and Todor Milanov. W-constraints for the total descendant potential of a simple singularity. Compos. Math., 149(5):840–888, 2013. [BLL+15] Christopher Beem, Madalena Lemos, Pedro Liendo, Wolfger Peelaers, Leonardo Rastelli, and Balt C. van Rees. Infinite chiral symmetry in four dimensions. Comm. Math. Phys., 336(3):1359–1433, 2015. [BPRvR15] Christopher Beem, Wolfger Peelaers, Leonardo Rastelli, and Balt C. van Rees. Chiral algebras of class S. J. High Energy Phys., (5):020, front matter+67, 2015. [BR17] Christopher Beem and Leonardo Rastelli. Vertex operator algebras, Higgs branches, and modular differential equations. arXiv:1707.07679[hep-th]. [BD04] Alexander Beilinson and Vladimir Drinfeld. Chiral algebras, volume 51 of Ameri- can Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004. [BFM] Alexander Beilinson, Boris Feigin, and Barry Mazur. Introduction to algebraic field theory on curves. preprint. [Bor86] Richard E. Borcherds. Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A., 83(10):3068–3071, 1986. [BS95] Peter Bouwknegt and Kareljan Schoutens, editors. W-symmetry, volume 22 of Ad- vanced Series in Mathematical Physics. World Scientific Publishing Co. Inc., River Edge, NJ, 1995. [BFN16a] Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. Instanton moduli spaces and W-algebras. Astérisque, (385):vii+128, 2016. [BFN16b] Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. Towards a mathematical definition of Coulomb branches of 3-dimensional N = 2 gauge theories, II arXiv:1601.03586 [math.RT]. [BFN17] Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. Ring objects in the equivariant derived satake category arising from coulomb branche. arXiv:1706.02112 math.RT]. [BG03] Kenneth A. Brown and Iain Gordon. Poisson orders, symplectic reflection algebras and representation theory. J. Reine Angew. Math., 559:193–216, 2003. [BK08] Jonathan Brundan and Alexander Kleshchev. Representations of shifted Yangians and finite W -algebras. Mem. Amer. Math. Soc., 196(918):viii+107, 2008. [CS16] Clay Córdova and Shu-Heng Shao. Schur indices, BPS particles, and Argyres-Douglas theories. J. High Energy Phys., (1):040, front matter+37, 2016. REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE 13

[DLN14] Chongying Dong, Xingjun Lin, Unitary vertex operator algebras J. Algebra, Vol. 397, 1 252–277. [DLN15] Chongying Dong, Xingjun Lin, and Siu-Hung Ng. Congruence property in conformal field theory. Algebra Number Theory, 9(9):2121–2166, 2015. [DM06] Chongying Dong and Geoffrey Mason. Integrability of C2-cofinite vertex operator algebras. Int. Math. Res. Not., pages Art. ID 80468, 15, 2006. [DS84] Vladimir G. Drinfeld´ and Vladimir V. Sokolov. Lie algebras and equations of Korteweg-de Vries type. In Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, pages 81–180. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984. [DSK06] Alberto De Sole and Victor G. Kac. Finite vs affine W -algebras. Japan. J. Math., 1(1):137–261, 2006. [DSKV13] Alberto De Sole, Victor G. Kac, and Daniele Valeri. Classical W-algebras and gener- alized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras. Comm. Math. Phys., 323(2):663–711, 2013. [ES10] Pavel Etingof and Travis Schedler. Poisson traces and D-modules on Poisson varieties. Geom. Funct. Anal., 20(4):958–987, 2010. With an appendix by Ivan Losev. [FL88] Vladimir A. Fateev and Sergei L. Lykyanov. The models of two-dimensional conformal quantum field theory with Zn symmetry. Internat. J. Modern Phys. A, 3(2):507–520, 1988. [Fe˘ı84] Boris L. Fe˘ıgin.Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras. Uspekhi Mat. Nauk, 39(2(236)):195–196, 1984. [FF84] Boris L. Fe˘ıginand Dmitry B. Fuchs. Verma modules over the Virasoro algebra. In Topology (Leningrad, 1982), volume 1060 of Lecture Notes in Math., pages 230–245. Springer, Berlin, 1984. [FF90] Boris Feigin and Edward Frenkel. Quantization of the Drinfel′d-Sokolov reduction. Phys. Lett. B, 246(1-2):75–81, 1990. [FF92] Boris Feigin and Edward Frenkel. Affine Kac-Moody algebras at the critical level and Gel´fand-Diki˘ıalgebras. In Infinite analysis, Part A, B (Kyoto, 1991), volume 16 of Adv. Ser. Math. Phys., pages 197–215. World Sci. Publ., River Edge, NJ, 1992. [Fre07] Edward Frenkel. Langlands correspondence for loop groups, volume 103 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007. [FBZ04] Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves, volume 88 of Mathematical Surveys and Monographs. American Mathematical Society, Provi- dence, RI, second edition, 2004. [FG09] Edward Frenkel and Dennis Gaitsgory. Localization of g-modules on the affine Grass- mannian. Ann. of Math. (2), 170(3):1339–1381, 2009. [FG10] Edward Frenkel and Dennis Gaitsgory. Weyl modules and opers without monodromy. In Arithmetic and geometry around quantization, volume 279 of Progr. Math., pages 101–121. BirkhäuserBoston Inc., Boston, MA, 2010. [FKW92] Edward Frenkel, Victor Kac, and Minoru Wakimoto. Characters and fusion rules for W -algebras via quantized Drinfel′d-Sokolov reduction. Comm. Math. Phys., 147(2):295–328, 1992. [FZ92] Igor B. Frenkel and Yongchang Zhu. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J., 66(1):123–168, 1992. [Gai12] Davide Gaiotto. N = 2 dualities. J. High Energy Phys., (8):034, front matter + 57, 2012. [GMN13] Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke. Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math., 234:239–403, 2013. [GG02] Wee Liang Gan and Victor Ginzburg. Quantization of Slodowy slices. Int. Math. Res. Not., (5):243–255, 2002. [GKO86] Peter Goddard and Adrian Kent and David Olive. Unitary representations of the Virasoro and super-Virasoro algebras. Comm. Math. Phys., 103(1):105–119, 1986. [Gin08] Victor Ginzburg. Harish-Chandra bimodules for quantized Slodowy slices Represent. Theory 13 (2009), 236-271. [Hua08] Yi-Zhi Huang. Rigidity and modularity of vertex tensor categories. Commun. Con- temp. Math., 10(suppl. 1):871–911, 2008. 14 TOMOYUKI ARAKAWA

[Kac98] Victor Kac. Vertex algebras for beginners, volume 10 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 1998. [Kac15] Victor Kac. Introduction to vertex algebras, Poisson vertex algebras, and integrable Hamiltonian PDE. arXiv:1512.00821[math-ph], to appear in in Perspectives in Lie Theory, Springer INdAM Series 19. [KRW03] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto. Quantum reduction for affine superalgebras. Comm. Math. Phys., 241(2-3):307–342, 2003. [KW88] Victor G. Kac and Minoru Wakimoto. Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Nat. Acad. Sci. U.S.A., 85(14):4956–4960, 1988. [KW89] Victor Kac and Minoru Wakimoto. Classification of modular invariant representations of affine algebras. In Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), volume 7 of Adv. Ser. Math. Phys., pages 138–177. World Sci. Publ., Teaneck, NJ, 1989. [KW90] Victor G. Kac and Minoru Wakimoto. Branching functions for winding subalgebras and tensor products. Acta Appl. Math., 21(1-2):3–39, 1990. [KW04] Victor G. Kac and Minoru Wakimoto. Quantum reduction and representation theory of superconformal algebras. Adv. Math., 185(2):400–458, 2004. [KW08] Victor G. Kac and Minoru Wakimoto. On rationality of W -algebras. Transform. Groups, 13(3-4):671–713, 2008. [KT00] Masaki Kashiwara and Toshiyuki Tanisaki. Characters of irreducible modules with non-critical highest weights over affine Lie algebras. In Representations and quanti- zations (Shanghai, 1998), pages 275–296. China High. Educ. Press, Beijing, 2000. [Li05] Haisheng Li. Abelianizing vertex algebras. Comm. Math. Phys., 259(2):391–411, 2005. [Los11] Ivan Losev. Finite-dimensional representations of W -algebras. Duke Math. J., Volume 159, Number 1 (2011), 99-143. [Lus91] George Lusztig. Intersection cohomology methods in representation theory. In Pro- ceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 155–174. Math. Soc. Japan, Tokyo, 1991. [MSV99] Fyodor Malikov, Vadim Schechtman, and Arkady Vaintrob. Chiral de Rham complex. Comm. Math. Phys., 204(2):439–473, 1999. [MO12] Davesh Maulik and Andrei Okounkov. Quantum Groups and Quantum Cohomology. ArXiv e-prints, (1211.1287), November 2012. [MT12] Gregory W. Moore and Yuji Tachikawa. On 2d TQFTs whose values are holomorphic symplectic varieties. In String-Math 2011, volume 85 of Proc. Sympos. Pure Math., pages 191–207. Amer. Math. Soc., Providence, RI, 2012. [Pre02] Alexander Premet. Special transverse slices and their enveloping algebras. Adv. Math., 170(1):1–55, 2002. With an appendix by Serge Skryabin. [SV13] Olivierivier Schiffmann and Eric Vasserot. Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A2. Publ. Math. Inst. Hautes Etudes´ Sci., 118:213–342, 2013. [SXY17] Jaewon Song, Dan Xie and and Wenbin Yan, Vertex operator algebras of Argyres- Douglas theories from M5-branes. arXiv:1706.01607[hep-th]. [Tac1] Yuji Tachikawa. On some conjectures on VOAs. http://member.ipmu.jp/yuji.tachikawa/transp/voaconj2.pdf. [Tac2] Yuji Tachikawa. On “categories’ of quantum field theories. in this volume. [Wan93] Weiqiang Wang. Rationality of Virasoro vertex operator algebras. Internat. Math. Res. Notices, (7):197–211, 1993. [XYY16] Dan Xie, Wenbin Yan, and Shing-Tung Yau. Chiral algebra of Argyres-Douglas theory from M5 brane. arXiv:1604.02155[hep-th]. [Zhu96] Yongchang Zhu. Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc., 9(1):237–302, 1996.