Parity and Spin CFT with boundaries and defects - I. Runkel L. Szegedy and G.M.T. Watts

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Parity and Spin CFT with boundaries and defects

Ingo Runkel 1, L´or´ant Szegedy 2, and G´erard M. T. Watts 3∗

1Fachbereich Mathematik, Universit¨at Hamburg,

Bundesstraße 55, 20146 Hamburg, Germany

2Faculty of Physics, University of Vienna,

Boltzmangasse 5, 1090 Wien, Austria

3Department of Mathematics, King’s College London,

Strand, London WC2R 2LS, UK

Abstract

This paper is a follow-up to [RWa] in which two-dimensional conformal ﬁeld theories in

the presence of spin structures are studied. In the present paper we deﬁne four types

of CFTs, distinguished by whether they need a spin structure or not in order to be

well-deﬁned, and whether their ﬁelds have parity or not. The cases of spin dependence

without parity, and of parity without the need of a spin structure, have not, to our

knowledge, been investigated in detail so far.

We analyse these theories by extending the description of CFT correlators via three-

dimensional topological ﬁeld theory developed in [FRS1] to include parity and spin.

In each of the four cases, the deﬁning data are a special Frobenius algebra

in a

suitable ribbon fusion category, such that the Nakayama automorphism of

is the

identity (oriented case) or squares to the identity (spin case). We use the TFT to deﬁne

correlators in terms of

and we show that these satisfy the relevant factorisation and

single-valuedness conditions.

We allow for world sheets with boundaries and topological line defects, and we specify

the categories of boundary labels and the fusion categories of line defect labels for each

of the four types.

The construction can be understood in terms of topological line defects as gauging

a possibly non-invertible symmetry. We analyse the case of a

-symmetry in some

detail and provide examples of all four types of CFT, with Bershadsky-Polyakov models

illustrating the two new types.

∗Emails: ingo.runkel@uni-hamburg.de ,lorant.szegedy@univie.ac.at ,gerard.watts@kcl.ac.uk

arXiv:2210.01057v2 [hep-th] 17 Jul 2023

Contents

1 Introduction and summary 3

1.1 Four types of conformal ﬁeld theory ................................ 3

1.2 Oriented parity CFT via topological ﬁeld theory ......................... 4

1.3 Spin CFT via oriented CFT with defects .............................. 6

1.4 Examples from a self-dual invertible object ............................ 8

1.5 Relation to gauging topological symmetries ............................ 10

2 Bordisms with boundaries, defects, and spin structures 13

2.1 Geometric description of open-closed spin surfaces ........................ 13

2.2 Combinatorial description of spin structures ............................ 18

2.3 Open-closed spin bordisms with defects .............................. 24

2.4 World sheets and bordisms ..................................... 25

3 Reshetikhin-Turaev TFT and conformal blocks 27

3.1 Modular categories .......................................... 27

3.2 Conformal blocks and 3d TFT ................................... 28

3.3 Compatibility with transport .................................... 30

4 3d TFTs with values in super-vector spaces 33

4.1 Super-vector spaces .......................................... 33

4.2 The trivial 3d TFT with values in SVect .............................. 34

4.3 Reshetikhin-Turaev TFT with values in SVect .......................... 36

4.4 Relation to conformal blocks of vertex operator super algebras ................. 37

5 Oriented CFT with parity signs 39

5.1 Algebras and modules ........................................ 39

5.2 Oriented CFT without boundaries and defects ........................... 42

5.3 Including boundaries and defects .................................. 46

5.4 Consistency of oriented correlators in the presence of parity ................... 50

5.5 Example: torus partition function .................................. 53

6 Spin CFT with parity signs 56

6.1 Z2-equivariant modules and bimodules ............................... 56

6.2 Spin CFT without defects and boundaries ............................. 58

6.3 Including boundaries and defects .................................. 64

6.4 Consistency of spin correlators ................................... 69

6.5 Example: torus partition functions for the four spin structures ................. 71

7 Examples 73

7.1 The trivial CFT and the Arf invariant ............................... 73

7.2 Computing multiplicity spaces for G≇1.............................. 75

7.3 Ising CFT and free fermions ..................................... 76

7.4 Three-state Potts and fermionic tetra-critical Ising CFTs ..................... 78

7.5 Bershadsky-Polyakov models: spin without parity and parity without spin ........... 81

A Appendix 86

A.1 Details for the combinatorial model of r-spin structures ..................... 86

A.2 Invariance of correlators under local moves ............................. 90

A.3 Proof of factorisation ......................................... 94

A.4 Algebras and multiplicity spaces for a self-dual invertible object ................. 102

A.5 Some details on the Bershadsky-Polyakov algebra at k=−1/2................. 105

1 Introduction and summary

The earliest and most thoroughly investigated two-dimensional conformal ﬁeld theories (CFTs) are

those where the world sheets are just complex one-dimensional manifolds, and where no further

geometric structure or parity grading is needed to deﬁne the theory. Such CFTs were the focus of the

foundational paper [BPZ] on rational conformal ﬁeld theory, and of early classiﬁcation results [CIZ].

These theories are algebraically well-understood via their relation to three-dimensional topological

ﬁeld theory [FRS1].

The second most studied kind of CFTs are those which require a spin structure to be well-deﬁned,

and where the state spaces are

-graded by “fermion number”. The most notable example is given

by massless free fermions. But already for this class of theories a systematic bootstrap formulation

in terms of crossing relations for operator product expansion coeﬃcients was not available until

recently [RWa]. In that paper, correlators of spin CFTs are described via topological line defects in

a “parity enhancement” of an underlying bosonic theory [NR2]. A related approach to fermionic

CFTs is via gauging suitable

-symmetries [KTT,HNT,GKu] giving a continuum version of the

Jordan-Wigner transformation. Minimal model CFTs with half-integer spin ﬁelds have also been

investigated earlier in [Pe,FGP], but without reference to spin structures.

In fact, the paper [RWa] and the present follow-up paper grew out of the desire to develop a systematic

approach to consistency conditions and their solutions for these “fermionic CFTs”. However, the

term “fermionic” can stand for diﬀerent properties of the theory: it can refer to the presence of

half-integer spin ﬁelds, i.e. ﬁelds that transform under the double cover of the rotation group

(2),

or it can refer to the statistics, i.e. to parity signs that arise when reordering the ﬁelds.

1.1 Four types of conformal ﬁeld theory

To illustrate this, let z(t), w(t) be two paths in the complex plane as follows: consider a correlator

z(t)

w(t)

z(0) = w(π)

z(π) = w(0)

Figure 1: The choice of path to interchange the positions of two identical ﬁelds

where two copies of a ﬁeld

are inserted, one at

(

) and one at

(

), plus possibly other ﬁelds

whose positions remain ﬁxed. Continue the correlator from

= 0 to

, so that the two copies of

ϕexchange places, as in Figure 1. In the theories we study in this paper there are three sources of

signs which can arise in this process,

ϕ(z(t))ϕ(w(t)) ···t=0 = (−1)M+P+Sϕ(z(t))ϕ(w(t)) ···t=π(1.1)

Here,

is the contribution from analytic continuation controlled by the conformal weights of the

ﬁelds,

gives the contribution of parity, and

arises from the eﬀect the monodromy has on the

spin structure. Single-valuedness requires (−1)M+P+S= 1.

It turns out that parity and spin-structure dependence can independently be present or not, leading

to four types of CFT:

type of CFT no parity parity

oriented 1

○2

○

spin 3

○4

○

(1.2)

Here, “oriented” refers to the fact that no spin structure is needed to obtain well-deﬁned correlators,

just the orientation induced by the complex structure on the world sheet. The CFTs mentioned in

the ﬁrst paragraph are of type

○

, in this case we have (

−

= 1 = (

−

for all ﬁelds. Those in

the second paragraph are of type

○

, where (

−

and (

−

take both values. In fact, the situation

is a little more subtle than (1.2), as we elaborate on in sections 1.2 and 1.3.

In this paper we present consistency conditions on correlators – namely compatibility with gluing

and monodromy-freeness with or without spin structure or parity – and a construction of solutions

to these conditions for all four types of CFTs. Our construction includes theories with boundaries

and topological line defects. We do this by extending the approach via 3d TFTs which was developed

for theories of type

○

in [FRS1] to the remaining three types. That is, we express the consistency

conditions and solutions in terms of a 3d TFT. For type

○

, the present paper provides the framework

to prove the claims in the prequel [RWa]; the proofs themselves will be presented in a further part of

this series.

Consistency conditions and their solutions for CFTs of types

○

and

○

have, to the best of our

knowledge, not been systematically studied in the literature so far.

1.2 Oriented parity CFT via topological ﬁeld theory

Let us describe our approach in more detail. The starting point is a rational vertex operator algebra

and the modular fusion category

formed by its representations. The category

deﬁnes the

Reshetikhin-Turaev TFT used in [FFFS,FRS1,FRS3,FjFRS1,FFS] to describe type 1

○theories.

To treat theories of type

○

, we slightly extend this TFT to include parity. Namely, let

be the

product b

C:= C⊠SVect , (1.3)

where

SVect

is the category of ﬁnite-dimensional super-vector spaces. Each simple object of

comes

in two variants in

, one parity even, and one parity odd. For two parity-odd objects, the braiding

acquires an additional minus sign. Note that

itself is not a modular fusion category as it has

symmetric centre SVect.

Let

Bord 3

(

) denote the category of three-dimensional bordisms with embedded

-decorated ribbon

graphs. We consider the TFT b

ZC:Bord 3(b

C)−→ SVect , (1.4)

which is basically the product of the Reshetikhin-Turaev TFT for

and the trivial

SVect

-valued

TFT, see Section 4for details.

Consider a surface Σ with marked points

p1, . . . , pn

labelled by

X1, . . . , Xn∈b

. To this, the TFT

assigns a vector space, which can be interpreted as the space of conformal blocks for the VOA

on Σ,

but where the

-representations are now of even or odd parity. This aﬀects the monodromy-behaviour

of conformal blocks by including parity signs.

From here on, the construction of CFT correlators is the same as in [FRS1], but we go through it in

some detail in Section 5to stress the eﬀect of including parity.

Let Σ be an oriented world sheet, possibly with boundaries and defect lines. Bulk insertions

are labelled by a tuple (

U, ¯

V, δ, ϕ

) and boundary ﬁelds by (

W, ν, ψ

), where

U, ¯

V∈ C

(not

) give

the holomorphic and antiholomorphic representation the bulk ﬁeld transforms in,

W∈ C

is the

representation for the boundary ﬁeld,

δ, ν ∈ {±

}

describe the parity of the ﬁelds, and

ϕ, ψ

take

values in appropriate multiplicity spaces which depend on the theory under consideration.

From Σ one constructs the double

= (Σ

⊔

rev

)

/∼

, where Σ

rev

is an orientation reversed copy of

Σ and

∼

identiﬁes boundary points of Σ and Σ

rev

. For example, if Σ is a disc, then

is a sphere. A

bulk insertion on Σ splits into two points on

, one on Σ labelled by

and one on Σ

rev

labelled by

. Boundary insertions result in a single marked point on

labelled by

. The ﬁrst key ingredient

of the TFT construction is:

The correlator for a world sheet Σ of an oriented parity CFT is an element of the space of

conformal blocks on the double e

Σ, i.e. in Bl(Σ) := b

ZC(e

Σ).

Specifying an oriented parity CFT now amounts to, ﬁrstly, giving the multiplicity spaces for bulk

ﬁelds in terms of

U, ¯

and the parity

(which may be zero-dimensional), and ditto for boundary

ﬁelds. This speciﬁes the ﬁeld content of the theory. Secondly, one has to give a collection of vectors

Corror

(Σ)

∈Bl

(Σ). The (bi)linear combinations of conformal blocks described by these vectors are

then the correlators of the oriented parity CFT.

The collection

{Corror

(Σ)

}Σ

has to satisfy two consistency conditions: it has to be monodromy free,

which amounts to mapping class group invariance, and it has to be compatible with gluing of world

sheets, see Section 5.4 for details. These consistency conditions can be expressed via the TFT b

ZC.

The second key point of the TFT construction is that consistent collections of correlators for CFTs

of type 2

○can be constructed from a suitable algebraic input:

From a symmetric special Frobenius algebra

B∈b

one can construct a consistent collection

of correlators

Corror

(Σ)

∈Bl

(Σ). Boundaries of Σ are labelled by

-modules and defect

lines by B-B-bimodules.

The correlators

Corror

(Σ) are described via the TFT

as follows: From Σ one obtains a bordism

MΣ:∅ −→ e

which contains a

-decorated ribbon graph deﬁned in terms of

and the ﬁeld insertions,

boundaries and line defects on Σ. As a 3-manifold,

MΣ

= Σ

[

−

/∼

, where (

z, t

)

∼

(

z, −t

) for

all z∈∂Σ, t∈[−1,1]. For example, if Σ is a disc, then MΣis a 3-ball. One then sets

Corror

B(Σ) := b

ZC(MΣ)∈Bl(Σ) .(1.5)

If in this construction one takes

B∈ C ⊂ b

, i.e.

is purely parity even, one recovers precisely the

construction in [FRS1] of correlators of oriented CFTs without parity, which is type

○

in the above

table.

A more intuitive way to understand this construction is to think of

MΣ

as a fattening of the world

sheet Σ, together with a surface defect inserted in place of the embedded copy of Σ in

MΣ

. The surface

defect determines how the holomorphic and antiholomorphic ﬁelds combine to form a consistent

collection of correlators. This interpretation of the construction in [FRS1] was given in [KS], and a

detailed study of surface defects in Reshetikhin-Turaev TFTs can be found in [FSV,CRS2]. We will,

however, not elaborate more on this point of view in the present paper.

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ParityandSpinCFTwithboundariesanddefectsIngoRunkel1,L´or´antSzegedy2,andG´erardM.T.Watts3∗1FachbereichMathematik,Universit¨atHamburg,Bundesstraße55,20146Hamburg,Germany2FacultyofPhysics,UniversityofVienna,Boltzmangasse5,1090Wien,Austria3DepartmentofMathematics,King’sCollegeLondon,Strand,LondonWC2R2L...

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Parity and Spin CFT with boundaries and defects - I. Runkel L. Szegedy and G.M.T. Watts

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