Vertex algebras with big center and a Kazhdan-Lusztig Correspondence Boris L. Feigin

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Vertex algebras with big center

and a Kazhdan-Lusztig Correspondence

Boris L. Feigin

Higher School of Economy,

Moscow, Russian Federation

The Hebrew University

Jerusalem, Israel

Simon D. Lentner

Universit¨at Hamburg,

Hamburg, Germany 1

Abstract

We study the semiclassical limit κ→ ∞ of the generalized quantum Langlands

kernel associated to a Lie algebra gand an integer level p. This vertex algebra acquires

a big center, containing the ring of functions over the space of g-connections. We

conjecture that the ﬁber over the zero connection is the Feigin-Tipunin vertex algebra,

whose category of representations should be equivalent to the small quantum group,

and that the other ﬁbres are precisely its twisted modules, and that the entire category

of representations is related to the quantum group with a big center. In this sense

we present a generalized Kazhdan-Lusztig conjecture, involving deformations by any

g-connection. We prove our conjectures in small cases (g,1) and (sl2,2) by explicitly

computing all vertex algebras and categories involved.

1Corresponding author: Bundesstraße 55, 20146 Hamburg, simon.lentner@uni-hamburg.de

arXiv:2210.13337v2 [hep-th] 19 Dec 2024

Contents

1 Introduction 2

1.1 Background ....................................... 2

1.2 Goals ........................................... 4

1.3 Content Part 1 ...................................... 7

1.4 Content Part 2 ...................................... 10

2 Twisted modules 13

2.1 G-twisted modules of ˆ

gκ................................ 15

2.2 SL2-twisted modules of the triplet algebra ...................... 17

2.3 Quantum groups ..................................... 24

2.4 Twisted free ﬁeld realization .............................. 26

3 Semiclassical limits and Sturm-Liouville operators 30

3.1 Limit of aﬃne Lie algebras: Connections ....................... 30

3.2 The Virasoro algebra .................................. 30

3.3 Limit of the Virasoro algebra .............................. 31

3.4 Sturm-Liouville operators ................................ 33

4 Vertex algebras with big center 37

4.1 Coupled vertex algebras ................................. 37

4.2 Limits of these vertex algebras ............................. 38

5 The case of simply-laced gat p= 1 39

5.1 The aﬃne coset ..................................... 39

5.2 The quantum Hamiltonian reduction ......................... 41

5.3 Representations and decomposition formulas ..................... 41

5.4 Bundle associated to the representations Vκ−1

µ⊗L1

ν................. 42

5.5 Bundle associated to the spectral ﬂow representations σℓ(Vκ−1

µ)⊗L1

ν....... 44

5.6 Bundle associated to the representations Tκ−1

µ,µ′⊗L1

ν................. 45

5.7 Bundle associated to the representations Wκ−1

µ⊗L1

ν, generic µ........... 45

5.8 A second limit and the aﬃne variant of the Feigin-Tipunin algebra ......... 46

6 Explicit computations for sl2, p = 1 47

7 Explicit computations for sl2, p = 2 49

7.1 The limit of the N=4 superconformal algebra ..................... 49

7.2 The limit of ˆ

osp(2|1)κ.................................. 49

7.3 The limit of the N=1 superconformal algebra and fermion ............. 54

7.4 The limit of the N=1 superconformal algebra and fermion II ............ 56

1 Introduction

1.1 Background

A vertex algebra Vis, roughly speaking, a complex-analytic version of a commutative

algebra, where the multiplication map depends on a formal complex variable

Y:V ⊗CV → V[[z, z−1]],

Here, V[[z, z−1]] denotes the Laurent series in a formal variable zwith coeﬃcients in V.

The axioms of a vertex operator algebra include a version of commutativity or locality,

which relates Y(a, z1)Y(b, z2) and Y(b, z2)Y(a, z1) up to formal delta-functions on the sin-

gularities z1, z2, z1−z2= 0. Another requirement is an action of the Virasoro algebra

on Vcompatible with conformal transformations of the variable z. Standard mathemat-

ical textbooks on vertex algebras include [Kac98,FBZ04]. Vertex algebras have a strong

relation to analysis, algebraic geometry and mathematical physics, in particular conformal

ﬁeld theory. A vertex algebra has a notion of representation, and under suitable conditions

the category of representations has the structure of a braided tensor category [HLZ].

A fundamental example is the vertex algebra Vκ(g) associated to an aﬃne Lie algebra

gκat some level κ, see Section 2.1. The conformal structure implies that certain analytic

functions attached to the vertex algebra solve the Knizhnik-Zamolochikov diﬀerential

equation. The solutions are multivalued around the singularities at zi=zj, and the

monodromy gives an action of the braid group on the space of solutions. The Drinfeld-

Kohno theorem [Koh88,Drin90], see e.g. [Kas95] Chp. 19, states that this braid group

action can be described by the braiding of the quantum group Uq(g), a deformation of the

universal enveloping algebra U(g) of a ﬁnite-dimensional semisimple Lie algebra g, which

has been deﬁned by Drinfeld for this very purpose. The Kazhdan-Lusztig correspondence

states that for generic κthe braided tensor category of suitable2representations of Vκ(g) is

equivalent to the category of weight modules of the quantum group Uq(g) at q=e

πi

m(κ+h∨)

with mthe lacing number of g[KL1,KL2,KL3,KL4] and [Zh08,Hu17]. Note that as

abelian categories, it is not diﬃcult so see that both categories are equivalent to the

category of integrable g-modules, independently of κ, as long as it is generic.

A closely related vertex algebra is the principal W-algebra Wκ(g) obtained from Vκ(g)

by a quantum Hamiltonian reduction, here with respect to the principal nilpotent orbit,

see [FBZ04] Chapter 15. In the smallest case g=sl2the W-algebra is the Virasoro algebra

itself, see Section 3.2. For results on the representation theory of W-algebras, see [Ara07].

A fundamental property of W-algebras is the Feigin-Frenkel duality [FF92], which states

an isomorphism Wκ(g)∼

=Wκ∗(g∨), where g∨is the Langlands dual and κ, κ∗are are pair

of dual levels fulﬁlling (κ+h∨)(κ∗+ (h∨)∨)m= 1, with (h∨)∨the dual Coxeter number

of the dual root system. This result and its limit κ→ −h∨, κ∗→ ∞ is of particular

importance for the geometric Langlands conjecture, which again draws from conformal

ﬁeld theory, as discussed in Frenkel’s lecture [Fr05].

A more recent example that has gathered considerable interest, is the Feigin-Tipunin

algebra Wp(g). It is deﬁned as a kernel of screenings and has a g∨-action with g∨-invariant

subalgebra Wκ(g); this is proven in [Sug21a] for simply-laced g, otherwise it is conjectured.

The conjectural logarithmic Kazhdan-Lusztig correspondence [FGST06,FT10,AM14,

Len21,Sug21a,Sug21b] states that Wp(g) has a ﬁnite non-semisimple category of rep-

resentation, which is equivalent as a braided tensor category to the representations of

the small quantum group uq(g), q2p= 1, a non-semisimple ﬁnite-dimensional Hopf alge-

bra quotient of Uq(g) for qa root of unity deﬁned by Lusztig [Lusz90], more precisely a

quasi-Hopf algebra variant ˜uq(sl2) [CGR20,GLO18,Ne21]. The smallest case Wp(sl2) has

been studied previously under the name triplet algebra, and in this case the conjecture

has been ﬁnally proven by the work of many authors [FGST06,AM08,NT09,TW13,

CGR20,MY20,CLR21,GN21]. For the respective statement for the singlet algebra and

a more systematic treatment of this and other cases see [CLR23]. The non-semisimplicity

2The Kazhdan-Lusztig category consists of representations of ˆ

gthat are smooth (for any vexists nwith

g≥nv= 0) and whose L0-eigenspaces are integrable with respect to the action of the horizontal algebra g

brings new challenges, see e.g. the surveys on logarithmic conformal ﬁeld theory [CR] and

the role of the coend [FS16].

Back to our example Vκ(g), it is intriguing to study limits where the parameter κ

tends to degenerate values. This depends of course on choices, more precisely a choice

of an integral form, which in most of our cases have been studied by [CL19] under the

name deformable family. For a certain integral form of Vκ(g) the limit κ→ ∞, which

physically corresponds to the semiclassical limit, leads to a commutative algebra, which

can be identiﬁed with the ring of functions on the space of regular g-valued connections,

as we review in Section 3.1. The limit of critical level κ→ −h∨acquires a big center,

which is the ring of functions on the space of g-opers, see [FBZ04] Proposition 16.8.4. By

[Ara11] this center is equal to the center of the reduction Wκ(g) at critical level. By the

Feigin-Frenkel duality mentioned above, this is equal to the center of the semiclassical

limit of Wκ(g∨). With sl∨

2=sl2this means that the semiclassical limit of the Virasoro

algebra is the ring of functions on the space of quadratic diﬀerentials resp. Sturm-Liouville

operators, and many properties of the associated diﬀerential equation are reﬂected in the

representation theory. We discuss this in more detail in Section 3.3 and 3.4.

1.2 Goals

The idea of the present article follows previous ideas in joint work of the ﬁrst author in

[FF92] and in [BBFLT13,BFL16,ACF22] and the line of work [CG17,CGL20,CDGG21,

CN22]: Suppose we have a deformable family of vertex algebras Vκ, which in the semiclas-

sical limit κ→ ∞ acquires a big central subalgebra Z(V∞), which coincides with the ring

of functions O(X) of some space X. The reader should have in mind for Xa Lie group.

Strictly speaking, in our case Xwill be the space of g-connections (resp. g-opers) on a

formal punctured disc, up to regular coordinate transformations G((t)), with composition

law(s) given by g-connections on a 3-punctured sphere with prescribed expansions around

the punctures. In the case of regular singularities such connections can be classiﬁed in

terms of their monodromy, which is an element in the Lie group G, and up to conjugation

in the Borel subgroup B. The existence and essentially uniqueness of a g-connection on a

3-punctured sphere with prescribed monodromies reproduces the group law in Gand B.

For such a vertex algebra V∞with big central subalgebra O(X), the vertex algebra

itself and its modules can be considered to be ﬁber bundles over X, where the ﬁbre V∞|x

at a point x∈Xis given by a quotient V∞/(Z−Z(x))Z∈Z(V∞). The ﬁbre at zero V∞|0

(resp. the identity in the Lie group) is again a vertex algebra, and all other ﬁbers are

modules over it. However, it is an important observation that for a vertex algebra with

a large central subalgebra, in contrast to an algebra with a large central subalgebra, the

other ﬁbres V∞|xare no proper modules over V∞|0, but rather twisted modules, meaning

that their vertex operator contains multivalued functions in the complex variable z, due to

the nontrivial braiding present. Spoken more categorically, the ﬁbres should be modules

in a X-graded tensor category with crossed braiding, see [ENOM09].

A diﬀerent, more familiar occurrence of G-crossed tensor categories are orbifolds: Let

Wbe a vertex algebra with a ﬁnite group or Lie group Gacting as automorphisms.

Then, again in general conjecturally, there is a G-crossed tensor category Lg∈GCgwith

C0the usual braided tensor category of representations of Wand with Cgthe C0-bimodule

category of g-twisted representations, which are representations of the vertex algebra W

involving multivalued analytic functions with monodromy controlled by g, see the initial

paragraphs of Section 2. The typical main application is that all of these representations

restrict to proper representations over the invariant subalgebra (or orbifold) WG, and

in good cases the representation category of WGconsists precisely of twisted modules

decomposed over this subalgebra. Categorically the decomposition is described by G-

equivariantization and again returns a proper braided tensor category.

Thirdly, a natural Hopf algebra with big center is the inﬁnite-dimensional quantum

group UKP dC

q(g) of Kac-Procesi-DeConcini [DKP92a]. This Hopf algebra has a central

subalgebra isomorphic to the ring of functions on the big cell or Poisson dual G∗resp.

(G∨)∗, and the quotient is the small quantum group uq(g). This is close, but not quite

right for our purposes: We now restrict ourselves to the Hopf algebra quotient with big

center O(B∨), whose category of representations ﬁbres over X=B∨, which is contained

in the Poisson dual as well. This is the abelian category for which we prove explicit corre-

spondences. In fact, it should be a B∨-crossed braided tensor category and the equivari-

antization should be given by the mixed quantum group [Gait21], see Question 1.3. If we

want to see a full G∨-graded category, we can note that any element in G∨is conjugate to

an element in B∨, and accordingly there should be a larger version of this quantum group

q(g) with central subalgebra O(G∨). Note that this matches the fact that g-connections

with regular singularities have associated monodromy in G, but up to conjugation in B.

The goal of this article is to present two related classes of examples Vκfor each (g, p),

where we can study the correspondence between ﬁbres in the limit, twisted representations

of the zero-ﬁbre, and corresponding representations of the quantum group with a big

center. In these examples, the general conjecture is that the limit V∞acquires as center the

ring of functions over g-connections resp. g-opers (for sl2Sturm-Liouville operators), and

such that the zero ﬁbre W=V∞|0is equal to the Feigin-Tipunin algebra Wp(g) [CN22].

On the other hand, the logarithmic Kazhdan-Lusztig conjecture relates this representation

theory to the small quantum group. Our ultimate goal would thus be four descriptions of

the same tensor category, extending the logarithmic Kazhdan-Lusztig correspondence:

A tensor category

with G∨-grading, G∨-action

and crossed braiding

g∈G∨Cg

modules over the

vertex algebra

with big center

Hκ∗

A(p)[g,∞]

modules over a

quantum group

with big center

q(g)

G∨-crossed extension

of modules over the

small quantum group

˜uq(g)

g-twisted modules

over the Feigin-

Tipunin vertex algebra

Wp(g)

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VertexalgebraswithbigcenterandaKazhdan-LusztigCorrespondenceBorisL.FeiginHigherSchoolofEconomy,Moscow,RussianFederationTheHebrewUniversityJerusalem,IsraelSimonD.LentnerUniversit¨atHamburg,Hamburg,Germany1AbstractWestudythesemiclassicallimitκ→∞ofthegeneralizedquantumLanglandskernelassociatedtoaLiealg...

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Vertex algebras with big center and a Kazhdan-Lusztig Correspondence Boris L. Feigin

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