P.I .C.
M . – 2018
Rio de Janeiro, Vol. 2 (1281–1300)

REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGSBRANCH CONJECTURE
T

A(荒川知幸)

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 [2002] as a natural quantizationof Slodowy slices. W -algebras play important roles not only in conformal field theories
but also in integrable systems (e.g. V. G. Drinfeld and Sokolov [1984], De Sole, V. G. Kac,
and Valeri [2013], and Bakalov and Milanov [2013]), the geometric Langlands program(e.g. E. Frenkel [2007], Gaitsgory [2016], Tan [2016], and Aganagic, E. Frenkel, and Okounkov [2017]) and four-dimensional gauge theories (e.g. Alday, Gaiotto, and Tachikawa[2010], Schiffmann and Vasserot [2013], Maulik and Okounkov [2012], and Braverman,
Finkelberg, and Nakajima [2016a]).
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 E. Frenkel, V. Kac, and Wakimoto [1992] that stated the existence and constructionof rational W -algebras, which generalizes the integrable representations of affine KacMoody 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.
MSC2010: primary 81R10; secondary 17B67, 17B67.

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TOMOYUKI ARAKAWA (荒川知幸)

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, andDan 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 j0i 2 V
and a vertex operation, which is a linear map V ˝V ! V ((z)), written a ˝b 7! a(z)b =P( n2Z a(n) z n 1 )b, such that the following are satisfied:• (Unit axioms) (j0i)(z) = 1V and a(z)j0i 2 a + zV [[z]] for all a 2 V .
• (Locality) (zw)n [a(z); b(w)] = 0 for a sufficiently large n for all a; b 2 V .

The operator T : a 7! a( 2) j0i 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 forany m; n 2 Z:
X m
(1)
[a(m) ; b(n) ] =(a(j ) b)(m+n j ) ;
j
j 0
 
X
m
(2)
(a(m) b)(n) =( 1)j(a(m j ) b(n+j ) ( 1)m b(m+n j ) a(j ) ):jj 0

By regarding the Borcherds identity as fundamental relations, representations of a vertex
algebra are naturally defined (see V. Kac [1998] and E. Frenkel and Ben-Zvi [2004] forthe 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 b
g = g[t; t 1 ] ˚ CK
be the affine Kac-Moody algebra associated with g. The commutation relations of b
g aregiven by(3) [xt m ; yt n ] = [x; y]t m+n + mım+n;0 (xjy)K;[K;bg] = 0(x; y 2 g; m; n 2 Z);

where ( j ) is the normalized invariant inner product of g, that is, ( j ) = 1/2h_ Killing
form and h_ is the dual Coxeter number of g. For k 2 C, let
V k (g) = U (bg) ˝U (g[t]˚CK) Ck ;

REPRESENTATION THEORY AND HIGGS BRANCH CONJECTURE

1283

where Ck is the one-dimensional representation of g[t] ˚ CK on which g[t ] acts triviallyand K acts as multiplication by k. There is a unique vertex algebra structure on V k (g)such that j0i = 1 ˝ 1 is the vacuum vector andX(xt n )z n 1 (x 2 g):x(z) =n2Z

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 )j0i. V k (g) is called the universal affine vertex algebra associated with g at a level k. The Borcherds identity (1) for x; y 2 g  V k (g) is identical tothe commutation relation (3) with K = k id, and hence, any V k (g)-module is a b
g-moduleof level k. Conversely, any smooth b
g-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 ofsmooth b
g-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 ab
g-module. The vertex algebra Lk (g) is called the simple affine vertex algebra associated with g at level k, and Lk (g) -Mod forms a full subcategory of V k (g) -Mod,the category of smooth b
g-modules of level k.

A vertex algebra V is called commutative if both sides of (1) are zero for all a; b 2 V ,m; n 2 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 (Borcherds [1986]).
Let X be an affine scheme, J1 X the arc space of X that is defined by the functorof points Hom(Spec R; J1 X) = Hom(Spec R[[t ]]; X ). The ring C[J1 X] is naturally adifferential algebra, and hence is a commutative vertex algebra. In the case that X is a
Poisson scheme C[J1 X] has the structure of Poisson vertex algebra (Arakawa [2012b]),which is a vertex algebra analogue of Poisson algebra (see E. Frenkel and Ben-Zvi [2004]
and V. Kac [2015] for the precise definition).
It is known by Haisheng Li [2005] that any vertex algebra V is canonically filtered, andhence can be regarded2 as a quantization of the associated graded Poisson vertex algebraL
gr V = p F p V /F p+1 V , where F  V is the canonical filtration of V . By definition,
F p V = spanC f(a1 )( n1 1) : : : (ar )( nr 1) j0i j ai 2 V; ni  0;Xini  pg:

1 However, the modules of a commutative vertex algebra are not the same as the modules as a differential

algebra.
2 This filtration is separated if V is non-negatively graded, which we assume.

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TOMOYUKI ARAKAWA (荒川知幸)

The subspace
RV := V /F 1 V = F 0 V /F 1 V  gr V
is called Zhu’s C2 -algebra of V . The Poisson vertex algebra structure of gr V restricts to
the Poisson algebra structure of RV , which is given byā:b̄ = a( 1) b;

fā; b̄g = a(0) b:

The Poisson variety
XV = Specm(RV )called the associated variety of V (Arakawa [2012b]). We have Li [2005] the inclusion
Specm(gr V )  J1 XV :(4)

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 (Arakawa [2012b]). 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∼

C[g ] ! RV ;x1 : : : xr 7! (x1 t 1 ) : : : (xr t 1 )j0i (xi 2 g):

Hence
(5)

XV k (g) Š g :Also, we have the isomorphism Spec(gr V k (g)) Š J1 g . By (5), we have XLk (g)  g ,which is G-invariant and conic. It is known Dong and Mason [2006] that(6)

Lk (g) is lisse () Lk (g) is integrable as a b
g-module (() k 2 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 !, called the conformalP
vector, such that the corresponding field !(z) = n2Z Ln z n 2 satisfies the following
3
conditions. (1) [Lm ; Ln ] = (m n)Lm+n + m 12 m ım+n;0 c 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
L
algebra V we set V∆ = fv 2 V j L0 v = ∆V g, so that V = ∆ V∆ . The universal affine

REPRESENTATION THEORY AND HIGGS BRANCH CONJECTURE

1285

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 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V (q) = trV (q L0 c/24 )is well-defined.
L
For a conformal vertex algebra V = ∆ V∆ , one defines Zhu’s algebra I. B. Frenkel
and Zhu [1992] Zhu(V ) of V by
Zhu(V ) = V /V ı V; V ı V = spanC fa ı b j a; b 2 V g;
 P∆awhere a ı b = i0a(i 2) b for a 2 ∆a . Zhu(V ) is a unital associative algebra
i
 
P
∆a
by the multiplication a  b =a(i 1) b. There is a bijection between thei 0iisomorphism classes Irrep(V ) of simple positive energy representation of V and that ofsimple Zhu(V )-modules (I. B. Frenkel and Zhu [1992] and Zhu [1996]). The grading of V
gives a filtration on Zhu(V ) which makes it quasi-commutative, and there is a surjectivemap(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, thelisse condition implies that any simple V -module is a positive energy representation (Abe,Buhl, and Dong [2004]).
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 categoryof integrable representations of b
g at level k. A theorem of Y. Zhu [1996] states that if
V is a rational, lisse, Z0 -graded conformal vertex algebra such that V0 = Cj0i, thenthe character M (e 2 i  ) converges to a holomorphic functor on the upper half plane forany M 2 Irrep(V ). Moreover, the space spanned by the characters M (e 2 i  ), M 2
Irrep(V ), is invariant under the natural action of SL2 (Z). This theorem was strengthened
in Dong, Lin, and Ng [2015] to the fact that fM (e 2 i  ) j M 2 Irrep(V )g forms a vector
valued modular function by showing the congruence property. Furthermore, it has beenshown in Huang [2008] that the category of V -modules form a modular tensor category.

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TOMOYUKI ARAKAWA (荒川知幸)

3

W -algebras

W -algebras are defined by the method of the quantized Drinfeld-Sokolov reduction thatwas discovered by Feigin and E. Frenkel [1990]. In the most general definition of W algebras given by V. Kac, Roan, and Wakimoto [2003], W -algebras are associated withthe pair (g; f ) of a simple Lie algebra g and a nilpotent element f 2 g. The corresponding
W -algebra is a one-parameter family of vertex algebra denoted by W k (g; f ), k 2 C. By
definition,0
W k (g; f ) := HDS;f(V k (g));
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 asfollows. Let fe; h; f g be an sl2 -triple associated with f , gj = fx 2 g j [h; x] = 2j g, so
L
LL
that g = j 2 1 Z gj . Set g1 = j 1 gj , g>0 = j 1/2 gj . Then  : g1 [t; t 1 ] !
2C, xt n 7! ın; 1 (f jx), defines a character. Let F = U (g>0 [t; t 1 ])˝U (g>0 [t ]+g1 [t;t 1 ])C , where C is the one-dimensional representation of g>0 [t] + g1 [t; t 1 ] on whichg1 [t; t 1 ] acts by the character  and g>0 [t] acts triviality. Then, for a V k (g)-module
M,
1


HDS;f
(M ) = H 2 + (g>0 [t; t 1 ]; M ˝ F );1

where H 2 + (g>0 [t; t 1 ]; N ) is the semi-infinite g>0 [t; t 1 ]-cohomology Feĭgin [1984]
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 E. Frenkel and Ben-Zvi [2004] and V. G. Kac and Wakimoto [2004], weiknow that HDS;f(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 Sf = f + ge  g Š g , the Slodowy slice at f , where ge denotes the centralizerof e in g. The affine variety Sf has a Poisson structure obtained from that of g byHamiltonian reduction (Gan and Ginzburg [2002]). We have(8)

XW k (g;f ) Š Sf ;Spec(gr W k (g; f )) Š J1 Sf(De Sole and V. G. Kac [2006] and Arakawa [2015a]). Also, we have
Zhu(W k (g; f )) Š U (g; f )(Arakawa [2007] and De Sole and V. G. Kac [2006]), where U (g; f ) is the finite W algebra associated with (g; f ) (Premet [2002]). Therefore, the W -algebra W k (g; f ) canbe regarded as an affinization of the finite W -algebra U (g; f ). The map (7) for W k (g; f )REPRESENTATION THEORY AND HIGGS BRANCH CONJECTURE

1287

is an isomorphism, which recovers the fact Premet [2002] and Gan and Ginzburg [2002]
that U (g; f ) is a quantization of the Slodowy slice Sf . The definition of W k (g; f ) naturally extends V. Kac, Roan, and Wakimoto [2003] 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 ) Š W k (g; f 0 ) if f and f 0 belong to the same nilpotent orbit of g.
The W -algebra associated with a minimal nilpotent element fmi m and a principal nilpotent
element fpri n 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 ofcentral charge 1 6(k 1)2 /(k+2) provided that k ¤ 2. In V. Kac, Roan, and Wakimoto[ibid.] it was shown that almost every superconformal algebra appears as the minimal W algebra W k (g; fmi n ) for some Lie superalgebra g, by describing the generators and therelations (OPEs) of minimal W -algebras. Except for some special cases, the presentationof 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 Bouwknegt
and Schoutens [1995]). In the case that g = sln , the non-critical principal W -algebras
is isomorphic to the Fateev-Lukyanov’s Wn -algebra Fateev and Lykyanov [1988] (Feigin
and E. Frenkel [1990] and E. Frenkel and Ben-Zvi [2004]). The critical principal W _g) of bg, that is thealgebra W h (g; fpri n ) is isomorphic to the Feigin-Frenkel center z(b_center of the critical affine vertex algebra V h (g) (Feigin and E. Frenkel [1992]). For ageneral f , we have the isomorphism_z(bg) Š Z(W h (g; f ));_(Arakawa [2012a] and Arakawa and Moreau [2018a]), 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 Arakawa and Premet [2017].

4

Representation theory of W -algebras

The definition of W k (g; f ) by the quantized Drinfeld-Sokolov reduction gives rise to afunctorV k (g) -Mod ! W k (g; f ) -Mod0
M 7! HDS;f(M ):Let Ok be the category O of b
g at level k. Then Ok is naturally considered as a full subcategory of V k (g) -Mod. For a weight  of bg of level k, let L() be the irreducible highest
weight representations of b
g with highest weight .

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TOMOYUKI ARAKAWA (荒川知幸)

Theorem 1 (Arakawa [2005]). Let fmi n 2 Omi n and let k be an arbitrary complex
number.
ii. We have HDS;f(M ) = 0 for any M 2 Ok and i 2 Znf0g. Therefore, the functor
mi n

Ok ! W k (g; fmi n ) -Mod;0
M 7! HDS;f
(M )
mi n

is exact.
0ii. For a weight  of bg of level k, HDS;f(L()) is zero or isomorphic to an irremi nducible highest weight representation of W k (g; fmi n ). Moreover, any irreducible
highest weight representation the minimal W -algebra W k (g; fmi n ) arises in thisway.
0
By Theorem 1 and the Euler-Poincaré principal, the character ch HDS;f(L()) is expressed in terms of ch L(). Since ch L() is known Kashiwara and Tanisaki [2000] forall non-critical weight , Theorem 1 determines the character of all non-critical irreducible
highest weight representation of W k (g; fmi n ). In the case that k is critical the characterof irreducible highest weight representation of W k (g; fmi n ) is determined by the LusztigFeigin-Frenkel conjecture (Lusztig [1991], Arakawa and Fiebig [2012], and E. Frenkel
and Gaitsgory [2009]).

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 representationsof superconformal algebras that appear as W k (g; fmi n ) once the character of irreducible
highest weight representations of b
g is given.
Let KLk be the full subcategory of Ok consisting of objects on which g[t ] acts locallyfinitely. Altough the functor(9)

Ok ! W k (g; f ) -Mod;0
M 7! HDS;f(M )is not exact for a general nilpotent element f , we have the following result.
i
Theorem 3 (Arakawa [2015a]). Let f , k be arbitrary. We have HDS;f(M ) = 0 for
mi n
any M 2 KLk and i ¤ 0. Therefore, the functor

KLk ! W k (g; fmi n ) -Mod;0
M 7! HDS;f
(M )
mi n

is exact.
In the case that f is a principal nilpotent element, Theorem 3 has been proved in E.
Frenkel and Gaitsgory [2010] using Theorem 4 below.

REPRESENTATION THEORY AND HIGGS BRANCH CONJECTURE

1289

The restriction of the quantized Drinfeld-Sokolv reduction functor to KLk does notproduce all the irreducible highest weight representations of W k (g; f ). However, one
can modify the functor (9) to the “ ”-reduction functor H 0;f (?) (defined in E. Frenkel,
V. Kac, and Wakimoto [1992]) to obtain the following result for the principal W -algebras.
Theorem 4 (Arakawa [2007]). Let f be a principal nilpotent element, and let k be an
arbitrary complex number.
i. We have H i ;fpri n (M ) = 0 for any M 2 Ok and i 2 Znf0g. Therefore, the functor
Ok ! W k (g; fpri n ) -Mod;

M 7! H 0;fpri n (M )is exact.
ii. For a weight  of bg of level k, H 0;fpri n (L()) is zero or isomorphic to an irreducible highest weight representation of W k (g; fpri n ). Moreover, any irreducible
highest weight representation the principal W -algebra W k (g; fpri n ) arises in thisway.

In type A we can derive the similar result as Theorem 4 for any nilpotent element f
using the work of Brundan and Kleshchev [2008] on the representation theory of finite
W-algebras (Arakawa [2011]). In particular the character of all ordinary irreducible representations of W k (sln ; f ) has been determined for a non-critical k.

5

BRST reduction of associated varieties

Let Wk (g; f ) be the unique simple graded quotient of W k (g; f ). The associated variety
XWk (g;f ) is a subvariety of XW k (g;f ) = Sf , which is invariant under the natural C  action on Sf that contracts to the point f 2 Sf . Therefore Wk (g; f ) is lisse if and only
if XWk (g;f ) = ff g.
0
By Theorem 3, Wk (g; f ) is a quotient of the vertex algebra HDS;f(Lk (g)), provided
that it is nonzero.
Theorem 5 (Arakawa [2015a]). For any f 2 g and k 2 C we have
XH 0

DS;f (Lk (g))Š XLk (g) \ Sf :Therefore,
0i. HDS;f(Lk (g)) ¤ 0 if and only if G:f  XLk (g) ;1290

TOMOYUKI ARAKAWA (荒川知幸)

0ii. If XLk (g) = G:f then XH 0 (Lk (g)) = ff g. Hence HDS;f(Lk (g)) is lisse, andDS;fthus, so is its quotient Wk (g; f ).

Theorem 5 can be regarded as a vertex algebra analogue of the corresponding result
Losev [2011] and Ginzburg [2009] for finite W -algebras.
0Note that if Lk (g) is integrable we have HDS;f(Lk (g)) = 0 by (6). Therefore we needto study more general representations of bg to obtain lisse W -algebras using Theorem 5.
Recall that the irreducible highest weight representation L() of b
g is called admissible
V. G. Kac and Wakimoto [1989] (1) if  is regular dominant, that is, h + ; ˛ _ i 62 Z0rerefor any ˛ 2 ∆reis the set of real roots of bg, ∆re+ , and (2) Q∆() = Q∆ . Here ∆+re_the set of positive real roots of bg, and ∆() = f˛ 2 ∆ j h + ; ˛ i 2 Zg, the set of
integral roots of . Admissible representations are (conjecturally all) modular invariant
representations of bg, that is, the characters of admissible representations are invariant under the natural action of SL2 (Z) (V. G. Kac and Wakimoto [1988]). The simple affinevertex algebra Lk (g) is admissible as a b
g-module if and only if(
h_ if (q; r _ ) = 1;p_
(10)
k + h = ; p; q 2 N; (p; q) = 1; p qhif (q; r _ ) = r _(V. G. Kac and Wakimoto [2008]). Here h is the Coxeter number of g and r _ is the lacityof g. If this is the case k is called an admissible number for bg and Lk (g) is called anadmissible affine vertex algebra.
Theorem 6 (Arakawa [2015a]). Let Lk (g) be an admissible affine vertex algebra.
i. (Feigin-Frenkel conjectrue) We have XLk (g)  N, the nilpotent cone of g.
ii. The variety XLk (g) is irreducible. That is, there exists a nilpotent orbit Ok of gsuch that
XLk (g) = Ok :iii. More precisely, let k be an admissible number of the form (10). Then(fx 2 g j (ad x)2q = 0g
if (q; r _ ) = 1;
XLk (g) =_fx 2 g j s (x)2q/r = 0g if (q; r _ ) = r _ ;where s is the highest short root of g and s is the irreducible finite-dimensional
representation of g with highest weight s .
From Theorem 5 and Theorem 6 we immediately obtain the following assertion, whichwas (essentially) conjectured by V. G. Kac and Wakimoto [2008].

REPRESENTATION THEORY AND HIGGS BRANCH CONJECTURE

1291

Theorem 7 (Arakawa [2015a]). Let Lk (g) be an admissible affine vertex algebra, andlet f 2 Ok . Then the simple affine W -algebra Wk (g; f ) is lisse.
In the case that XLk (g) = G:fpri n , the lisse W -algebras obtained in Theorem 7 is theminimal series principal W -algebras studied in E. Frenkel, V. Kac, and Wakimoto [1992].
In the case that g = sl2 , these are exactly the minimal series Virasoro vertex algebras(Feĭgin and Fuchs [1984], Beilinson, Belavin, V. Drinfeld, and et al. [2004], and Wang
[1993]). 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 (V. G. Kac and Wakimoto [2008] and Arakawa[2015a]).

6

The rationality of minimal series principal W-algerbas

An admissible affine vertex algebra Lk (g) is called non-degenearte (E. Frenkel, V. Kac,
and Wakimoto [1992]) if
XLk (g) = N = G:fpri n :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, anadmissible number k of the form (10) is non-degenerate if and only if(hif (q; r _ ) = 1;q  _L _r h
if (q; r _ ) = r _where L h_ is the dual Coxeter number of the Langlands dual Lie algebra L g. For a nondegenerate admissible number k, the simple principal W -algebra Wk (g; fpri n ) is lisse by
Theorem 7.
The following assertion settles the Frenkel-Kac-Waimoto conjecture E. Frenkel, V.
Kac, and Wakimoto [ibid.] in full generality.
Theorem 8 (Arakawa [2015b]). Let k be a non-degenerate admissible number. Then thesimple principal W -algebra Wk (g; fpri n ) is rational.
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 [1995].
Theorem 9 (Arakawa [2016]). 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.

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TOMOYUKI ARAKAWA (荒川知幸)

The following assertion, which has been widely believed since V. G. Kac and Wakimoto[1990] and E. Frenkel, V. Kac, and Wakimoto [1992], gives a yet another realization ofminimal series principal W -algebras.
Theorem 10 (Arakawa, Creutzig, and Linshaw [2018]). Let g be simply laced. For anadmissible affine vertex algebra Lk (g), the vertex algebra (Lk (g) ˝ L1 (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 = sl2 and k is a non-negative integer, the statement of Theorem 10 iswell-known as the GKO construction of the discrete series of the Virasoro vertex algebras
Goddard, Kent, and Olive [1986]. Some partial results have been obtained previously inArakawa, Lam, and Yamada [2017] and Arakawa and Jiang [2018]. From Theorem 10,
it follows that the minimal series principal W -algebra Wp/q h_ (g; fpri n ) of ADE typeis unitary, that is, any simple Wp/q h_ (g; fpri n )-module is unitary in the sense of Dongand Lin [2014], if and only if jp qj = 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 Beem, Lemos, Liendo, Peelaers, Rastelli,and van Rees [2015] have constructed a remarkable map(11)Φ : f4d N = 2 SCFTsg ! fvertex algebrasg

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 Beem,
Lemos, Liendo, Peelaers, Rastelli, and van Rees [ibid.], we havec2d =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 affinevertex algebras never appear by this correspondence.
The main examples of vertex algebras considered in Beem, Lemos, Liendo, Peelaers,
Rastelli, and van Rees [ibid.] are the affine vertex algebras Lk (g) of types D4 , F4 , E6 , E7 ,

REPRESENTATION THEORY AND HIGGS BRANCH CONJECTURE

1293

E8 at level k = h_ /6 1, which are non-rational, non-admissible affine vertex algebras
studies in Arakawa and Moreau [2016]. One can find more examples in the literature, seee.g. Beem, Peelaers, Rastelli, and van Rees [2015], Xie, Yan, and Yau [2016], Córdova
and Shao [2016], and Song, Xie, and Yan [2017].
Now, there is another important invariant of a 4d N = 2 SCFT T , called the Higgs
branch, which we denote by H iggsT . The Higgs branch H iggsT is an affine algebraic
variety that has the hyperKähler structure in its smooth part. In particular, H iggsT is a(possibly singular) symplectic variety.
Let T be one of the 4d N = 2 SCFTs studied in Beem, Lemos, Liendo, Peelaers,
Rastelli, and van Rees [2015] such that that Φ(T ) = Lk (g) with k = h_ /6 1 for types
D4 , F4 , E6 , E7 , E8 as above. It is known that H iggsT = Omi n , and this equals Arakawa
and Moreau [2016] to the the associated variety XΦ(T ) . It is expected that this is not justa coincidence.
Conjecture 11 (Beem and Rastelli [2017]). For a 4d N = 2 SCFT T , we have
H iggsT = 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 amathematical 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 branchesof three-dimensional N = 4 gauge theories whose mathematical definition has been given
by Braverman, Finkelberg, and Nakajima [2016b].
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(Arakawa and Kawasetsu [n.d.]) if XV has only finitely many symplectic leaves. If thisis the case symplectic leaves in XV are algebraic (Brown and Gordon [2003]). Clearly,
lisse vertex algebras are quasi-lisse. The simple affine vertex algebra Lk (g) is quasi-lisse
if and only if XLk (g)  N. In particular, admissible affine vertex algebras are quasi-lisse.
See Arakawa and Moreau [2016, 2017, 2018b] 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 Zhu [1996] using a theorem of Etingof and Schelder
Etingof and Schedler [2010], we obtain the following assertion.
Theorem 12 (Arakawa and Kawasetsu [n.d.]). Let V be a quasi-lisse Z0 -graded conformal vertex algebra such that V0 = C. Then there only finitely many simple ordinary

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TOMOYUKI ARAKAWA (荒川知幸)

V -modules. Moreover, for an ordinary V -module M , the character 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 SL2 (Z), Theorem 12 implies that a quasi-lisse vertex algebra possesses acertain 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. Thisis something that has been conjectured by physicists (Beem and Rastelli [2017]).
There is a distinct class of four-dimensional N = 2 superconformal field theories called
the theory of class S (Gaiotto [2012] and Gaiotto, Moore, and Neitzke [2013]), where S
stands for 6. The vertex algebras obtained from the theory of class S is called the chiral algebras of class S (Beem, Peelaers, Rastelli, and van Rees [2015]). The Moore-Tachikawa
conjecture Moore and Tachikawa [2012], which was recently proved in Braverman, Finkelberg, and Nakajima [2017], describes the Higgs branches of the theory of class S in termsof two-dimensional topological quantum field theories.
Let V be the category of vertex algebras, whose objects are semisimple groups, andHom(G1 ; G2 ) is the isomorphism classes of conformal vertex algebras V with a vertex
algebra homomorphismVh_1(g1 ) ˝ Vh_2(g2 ) ! Vsuch that the action of g1 [t ] ˚ g2 [t] on V is locally finite. Here gi = Lie(Gi ) and h_iis the dual Coxeter number of gi in the case that gi is simple. If gi is not simple we_understand V hi (gi ) to be the tensor product of the critical level universal affine vertex
algebras corresponding to all simple components of gi . The composition V1 ı V2 of V1 2
Hom(G1 ; G2 ) and V2 2 Hom(G1 ; G2 ) is given by the relative semi-infinite cohomology
1V1 ı V2 = H 2 + (bg2 ; g2 ; V1 ˝ V2 );where bg2 denotes the direct sum of the affine Kac-Moody algebra associated with thesimple components of g2 . By a result of Arkhipov and Gaitsgory [2002], one finds that thechidentity morphism idG is the algebra DG
of chiral differential operators on G (Malikov,
Schechtman, and Vaintrob [1999] and Beilinson and V. Drinfeld [2004]) at the criticallevel, whose associated variety is canonically isomorphic to T  G.
The following theorem, which was conjectured in Beem, Lemos, Liendo, Peelaers,
Rastelli, and van Rees [2015] (see Tachikawa [n.d.(a),(b)]) for mathematical expositions),
describes the chiral algebras of class S.
Theorem 13 (Arakawa [n.d.]). Let B2 the category of 2-bordisms. There exists a uniquemonoidal functor G : B2 ! V which sends (1) the object S 1 to G, (2) the cylinder,REPRESENTATION THEORY AND HIGGS BRANCH CONJECTURE

1295

ch
which is the identity morphism idS 1 , to the identity morphism idG = DG
, and (3) the
0
ch
BF N
cap to HDS;fpri n (DG ). Moreover, we have XG (B) Š G (B) for any 2-bordism B,

N
where BF
is the functor form B2 to the category of symplectic variaties constructed inG
Braverman, Finkelberg, and Nakajima [2017].

The last assertion of the above theorem confirms the Higgs branch conjecture for thetheory of class S.

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Received 2017-12-06.
TA(荒川知幸)
arakawa@kurims.kyoto-u.ac.jp