THREE DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORY FROMUqgl1j1ANDU1j1CHERNSIMONS THEORY NATHAN GEER AND MATTHEW B. YOUNG

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THREE DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORY

FROM Uq(gl(1|1)) AND U(1|1) CHERN–SIMONS THEORY

NATHAN GEER AND MATTHEW B. YOUNG

Abstract.

We introduce an unrolled quantization

(

1)) of the complex Lie superalgebra

1) and use its categories of weight modules to construct and study new three dimensional

non-semisimple topological quantum ﬁeld theories. These theories are deﬁned on categories of

cobordisms which are decorated by ribbon graphs and cohomology classes and take values in

categories of graded super vector spaces. Computations in these theories are enabled by a detailed

study of the representation theory of

(

1)), both for generic and root of unity

. We argue

that by restricting to subcategories of integral weight modules we obtain topological quantum

ﬁeld theories which are mathematical models of Chern–Simons theories with gauge supergroups

psl

1) and

1) coupled to background ﬂat

C×

-connections, as studied in the physics literature

by Rozansky–Saleur and Mikhaylov. In particular, we match Verlinde formulae and mapping class

group actions on state spaces of non-generic tori with results in the physics literature. We also

obtain explicit descriptions of state spaces of generic surfaces, including their graded dimensions,

which go beyond results in the physics literature.

Contents

Introduction 1

1. Preliminary material 7

2. Relative modular categories from UE

q(gl(1|1)) 17

3. A TQFT for arbitrary q31

4. A TQFT for root of unity q43

5. TQFTs from integral modules 49

6. Comparison with supergroup Chern–Simons and Wess–Zumino–Witten theories coupled

to background ﬂat C×-connections 49

References 52

Introduction

This paper constructs and studies new three dimensional topological quantum ﬁeld theories from

non-semisimple categories of representations of the unrolled quantum group of the complex Lie

superalgebra

1) and establishes a relationship between these theories and various supergroup

Chern–Simons theories studied in the physics literature. Before stating our results in more detail,

we provide some context.

Date: December 9, 2022.

2010 Mathematics Subject Classiﬁcation. Primary: 81T45; Secondary 20G42.

Key words and phrases. Topological quantum ﬁeld theory. Chern–Simons theory. Representation theory of

quantum supergroups.

arXiv:2210.04286v2 [math.QA] 7 Dec 2022

2 N. GEER AND M. B. YOUNG

Background and motivation.

Chern–Simons theory is a three dimensional quantum gauge

theory which was introduced by Witten to give a physical realization of the Jones polynomial

[

Wit89a

]. The input data is a compact Lie group

, the gauge group, and a class

k∈H4

(

;

the level, satisfying a non-degeneracy condition. At the physical level of rigor, Chern–Simons

theory produces invariants of links in closed oriented 3-manifolds which are local in the sense that

they can be computed using cutting and gluing techniques. Witten also argued that calculations

in Chern–Simons theory can be made using its boundary conformal ﬁeld theory, a Wess–Zumino–

Witten theory with target

, thereby importing techniques from the theories of rational vertex

operator algebras and aﬃne Lie algebras to knot theory and 3-manifold topology. Since the

physical deﬁnition of Chern–Simons theory relies on path integrals, it cannot, at present, be

used to give a mathematical construction of the theory. Motivated by this, Reshetikhin and

Turaev constructed from a modular tensor category

a three dimensional topological quantum

ﬁeld theory

CobC→

ectC

which, in particular, encodes invariants of

-colored links in

3-manifolds with the expected locality properties [

RT90

RT91

Tur94

]. When

is simple and

simply connected, in which case

is an integer, there is a modular tensor category

(

G, k

) of

semisimpliﬁed representations of the quantum group

(

) at a

-dependent root of unity

and

ZC(G,k)

is a mathematical model of Chern–Simons theory [

RT91

And92

TW93

Saw06

]. Crucial

to the construction of

is that modular tensor categories are semisimple, have only ﬁnitely many

isomorphism classes of simple objects and have the property that simple objects have non-zero

quantum dimension.

Extensions of Chern–Simons theory to more general classes of gauge groups have been proposed

in the physics literature. This includes gauge groups which are non-compact Lie groups and complex

reductive groups [Wit91, BNW91, Guk05, DGLZ09] and Lie supergroups [Wit89b, Hor90, RS92,

RS93

RS94

KS09

GW10

Mik15

MW15

]. Such extensions are expected to have applications

to many areas, including the Volume Conjecture, logarithmic conformal ﬁeld theory and three

dimensional quantum gravity. Mathematical constructions of these extensions have largely

been obstructed by technical and conceptual diﬃculties which appear when moving beyond

compact gauge groups. For example, the Chern–Simons/Wess–Zumino–Witten correspondence

is fundamentally unclear in these extensions, thereby preventing the use of recent advances in

the theory of logarithmic vertex operator algebras [

GQS07

QS07

CR13

CMY22

]. Since Chern–

Simons theories with non-compact gauge groups involve categories of line operators which are

non-semisimple, have inﬁnitely many isomorphism classes of simple objects and have simple objects

with vanishing quantum dimension, there are serious obstructions to applying the Reshetikhin–

Turaev construction.

It is therefore of interest to extend Reshetikhin–Turaev-type constructions beyond modular

tensor categories. Early approaches to such extensions are given in the works of Hennings [

Hen96

]

and Kerler and Lyubashenko [

KL01

]. More recently, the ﬁrst author and collaborators have created

a theory of renormalized quantum invariants of low dimensional manifolds [

GPMT09

CGPM14

BCGPM16

DRGPM20

DR22

]. Central categorical structures of this theory include relative

pre-modular categories, non-degenerate relative pre-modular categories and relative modular

categories which produce invariants of links, invariants of closed 3-manifolds and three dimensional

topological quantum ﬁeld theories, respectively. In this paper we focus on relative modular

categories, the strongest of these structures, which are generalizations of modular tensor categories

that allow for non-semisimplicity, inﬁnitely many simple objects and simple objects with vanishing

quantum dimension. Roughly speaking, a relative modular category

is a ribbon category which is

Uq(gl(1|1)) AND U(1|1) CHERN–SIMONS THEORY 3

compatibly graded by an abelian group

and carries a degree 0 monoidal action of a group Z. It is

required that there exists a suﬃciently small subset X

⊂G

such that the full subcategories

Cg⊂ C

g∈G\

X, are semisimple and have only ﬁnitely many isomorphism classes of simple objects modulo

Z. The associated three dimensional topological quantum ﬁeld theory

obad

C→

ectZ- gr

constructed by De Renzi [

DR22

], is deﬁned on a category of admissible decorated three dimensional

cobordisms and takes values in a braided monoidal category of Z-graded complex vector spaces.

When

is in fact a modular tensor category, the theory

reduces to that of Reshetikhin and

Turaev. In general,

enjoys many new features not shared by modular theories, including the

ability to distinguish homotopy classes of lens spaces and produce representations of mapping

class groups with interesting properties, such as having Dehn twists act with inﬁnite order.

Categories of weight modules over unrolled quantum groups of complex simple Lie algebras are

relative modular and their associated topological quantum ﬁeld theories have been the subject of

recent interest [

BCGPM16

DRGPM20

CDGG21

DR22

]. The case of unrolled quantum groups

of complex Lie superalgebras is more subtle. For example, depending on the precise class of weight

modules being considered, the Lie superalgebras

(

m|n

, produce categories which are

relative modular or only non-degenerate pre-relative modular [

Ha18

AGPM21

GPMR21

Ha22

The resulting topological quantum ﬁeld theories have not been studied. This paper presents the

ﬁrst systematic study of topological quantum ﬁeld theories arising from quantum supergroups

and suggests that examples arising from higher rank Lie superalgebras admit natural physical

realizations, in contrast to the original expectations of Mikhaylov and Witten [

MW15

]. Further

examples of (non-degenerate) relative pre-modular categories, some of which are conjectured

to extend to relative modular categories, and their applications to knot theory and 3-manifold

topology can be found in [GPM07, GPM10, AGPM21].

Main results.

We construct new examples of relative modular categories using the representation

theory of an unrolled quantization of the complex Lie superalgebra

1). We study in detail

the resulting topological quantum ﬁeld theories and connect them to

psl

1) and

1) Chern–

Simons theories and

1) Wess–Zumino–Witten theory, as studied in the physics literature by

Rozansky and Saleur [

RS92

RS93

RS94

] and Mikhaylov [

Mik15

]. We also connect our work to

mathematical results on the quantum topology of

1) [

FN91

KS91

Res92

Vir06

Sar15

BI22

In the remainder of this introduction we outline the structure of the paper and state the main

results.

We begin in Section 1 by establishing our conventions for relative modular categories and

recalling how these categories can be used to deﬁne invariants of links and 3-manifolds and,

ultimately, three dimensional topological quantum ﬁeld theories. Our ﬁrst main result asserts ﬁnite

dimensionality of the state spaces of these ﬁeld theories under the assumption that the input relative

modular category is TQFT ﬁnite in the sense of Deﬁnition 1.15. TQFT ﬁniteness is a relatively

weak condition and is straightforward to verify in concrete examples. For example, a relative

modular category which is locally ﬁnite abelian with ﬁnitely many projective indecomposable

objects modulo Zin each degree g∈Gis TQFT ﬁnite.

Theorem A

(Theorem 1.16)

Let

be a relative modular category which is TQFT ﬁnite. Then

for each decorated surface S ∈ Cobad

C, the state space ZC(S)∈VectZ- gr

Cis ﬁnite dimensional.

Theorem A provides a general reason for the observed ﬁnite dimensionality of state spaces in all

known examples, namely those arising from relative modular categories of modules over unrolled

quantum groups of complex simple Lie (super)algebras [

BCGPM16

DRGPM20

AGPM21

Ha22

]

4 N. GEER AND M. B. YOUNG

and those of this paper. Theorem A is proved by exhibiting an explicit, combinatorially deﬁned

spanning set of ZC(S) using special C-colorings of a ﬁxed spine of S.

In Section 2 we introduce a non-standard quantization

(

1)) of

1). The algebra

(

1)) is an unrolled version of standard quantizations of

1) [

Kul89

KT91

Res92

], in

the sense of [

CGPM15

]. Fix

q∈C\ {

,±

}

. The superalgebra

(

1)) is generated by even

Cartan generators

K±1

and odd Serre generators

. The generator

should be viewed

as a logarithm of

, but this relation is not imposed at the level of the algebra. Instead, we

consider the category

of all weight

(

1))-modules on which

acts by

. Let also

Dq,int ⊂ Dq

be the full subcategory of weight modules whose

-weights are integral; no integrality

-weights is assumed. A natural Hopf superalgebra structure on

(

1)) gives

and

Dq,int

the structure of rigid monoidal categories. We study

and

Dq,int

in detail, obtaining

complete descriptions of their simple and projective indecomposable objects. The culmination of

our results in Section 2 is summarized as follows.

Theorem B

(Theorems 2.16, 2.20, 2.22, 2.23, 2.25)

The categories

and

Dq,int

admit relative

modular structures which depend on whether or not

is a root of unity and, if so, the parity

of the order of the root of unity. In particular,

and

Dq,int

are generically semisimple ribbon

categories. Moreover,

and

Dq,int

are TQFT ﬁnite with respect to any of the above relative

modular structures.

More precisely,

and

Dq,int

admit relative modular structures for any

q∈C\ {

,±

}

, which

we refer to as the case of arbitrary

, and admit second, distinct, relative modular structures

when

is a root of unity. Denote by

either of the categories

and

Dq,int

with any of the

relative modular categories of Theorem B and by

obad

C→

ectZ- gr

the associated topological

quantum ﬁeld theory. In all cases, the braided category V

ectZ- gr

is a graded version of complex

super vector spaces.

In the remainder of the paper we study in detail

(Sections 3-5) and their relationship

psl

1) and

1) Chern–Simons and Wess–Zumino–Witten theories (Section 6). To avoid

cumbersome statements, in the introduction we state precise results only for

with

primitive

rth

root of unity,

r≥

3 odd. In this case, the category

is graded by

C/Z×C/Z

corresponding to (

E, G

)-weights modulo

Z×Z

, with X=

2Z/Z× {¯

and Z=

Z×Z×Z/

. In

the body of the paper we treat all cases of Theorem B.

Our ﬁrst series of results concerns the Z-graded vector space

(

) =

Lk∈ZZC,k

(

) assigned

to a decorated surface

S ∈

obad

. Part of the data of

is a cohomology class

ω∈H1

(

;

) on

the underlying closed surface

. The description of

(

) simpliﬁes considerably when

generic in the sense that ω(γ)∈G\Xfor some simple closed curve γ⊂ S0.

Theorem C

(Theorems 3.5 and 4.3)

Let

be a decorated connected surface of genus

g≥

without marked points and with generic cohomology class

. For any (

β, ¯

)

∈G

, the partition

function of

S×S1

(¯

β,¯

, the closed decorated 3-manifold obtained by crossing

with

and extending

ωto ω⊕(¯

β, ¯

b), is

ZDq(S × S1

(¯

β,¯

b))=(−1)g+1r2g−1

r−1

i=0

(q¯

β+i−q−¯

β−i)2g−2.

Uq(gl(1|1)) AND U(1|1) CHERN–SIMONS THEORY 5

Moreover, the partition function

ZDq

(

S × S1

(¯

β,¯

)and state space

ZDq

(

)are related through the

Verlinde formula

ZDq(S × S1

(¯

β,¯

b)) = X

(n,n0)∈Z2

χ(ZDq,(n,n0,•)(S))q−2r(βn0+bn),

where

(

ZDq,(n,n0,•)

(

)) denotes the Euler characteristic of the

-graded subspace of

ZDq

(

)

consisting of vectors with Z-degree of the form (n, n0,•).

Theorem C is proved using an explicit surgery presentation of trivial circle ﬁbrations and the

representation theoretic results of Section 2. The strategy of proof is similar to its counterpart for

topological quantum ﬁeld theories arising from the unrolled quantum group

(

(2)) [

BCGPM16

In the body of the paper we allow

to carry marked points, in which case

ZDq

(

S×S1

(¯

β,¯

) depends

also on ¯

To obtain a more detailed understanding of

(

), we ﬁrst prove in Theorems 3.3 and 4.1 that,

in the present class of examples and under the assumed genericity of

, the general spanning set

(

) constructed in Theorem A can be reduced to a much smaller set. Theorems 3.3 and

4.1 can be seen as vanishing results, asserting that

(

) is concentrated in a restricted set of

Z-degrees. Using these results, we prove that

lim

β→¯

4ZDq(S × S1

(¯

β,¯

b)) = dimCZDq(S).

Together with Theorem C, this leads to the following explicit description of state spaces of generic

surfaces.

Theorem D

(Corollaries 3.7 and 4.5)

Let

be a decorated connected surface of genus

g≥

without marked points and with generic cohomology class. Then

ZDq,−k

(

)is trivial unless

= (0

, d, ¯

)for some

d∈

[

−

(

g−

, g −

∩rZ

, in which case it is of dimension

r2g2g−2

g−1−|d|

. In

particular, the total dimension of ZDq(S)is

dimCZDq(S) = r2gbg−1

n0=−bg−1

rc2g−2

g−1− |n0|r.

When the cohomology class

is not generic, the vector space

(

) is considerably more

complicated. In this setting we restrict attention to the torus, where we again obtain a complete

description of

(

). We prove that

ZDq

(

) =

ZC,0

(

) is two dimensional for arbitrary

(Propositions 3.9 and 3.11) and that

ZDq

(

) =

ZC,0

(

) is

+ 1 dimensional for

a primitive

rth

root of unity (Proposition 4.6). The result for arbitrary

is particularly surprising since

its contrasts the conjectured behavior of TQFTs constructed from

(

(2)) [

BCGPM16

]. We

construct explicit bases of ZC(S) to prove the following result.

Theorem E

(Theorems 3.13 and 4.7)

Let

be a decorated connected surface of genus one

without marked points and with non-generic cohomology class

. The mapping class group action

)on

ZDq

(

)admits an explicit description which, in particular, shows that the Dehn

twist acts with inﬁnite order.

Mapping class group actions with properties similar to those of Theorem E for arbitrary

are

obtained using the representation theory of UH

q(sl(2)) at a root of unity in [BCGPM16].

For a given

, the theories

ZDq

and

ZDq,int

are closely related. A priori, the signiﬁcant diﬀerence

in the gradings of these categories- the grading group for

Dq,int

is much smaller than that of

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THREEDIMENSIONALTOPOLOGICALQUANTUMFIELDTHEORYFROMUq(gl(1j1))ANDU(1j1)CHERN{SIMONSTHEORYNATHANGEERANDMATTHEWB.YOUNGAbstract.WeintroduceanunrolledquantizationUEq(gl(1j1))ofthecomplexLiesuperalgebragl(1j1)anduseitscategoriesofweightmodulestoconstructandstudynewthreedimensionalnon-semisimpletopologicalq...

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THREE DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORY FROMUqgl1j1ANDU1j1CHERNSIMONS THEORY NATHAN GEER AND MATTHEW B. YOUNG

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