KIAS-P22065 On Classification of Fermionic Rational Conformal Field Theories

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KIAS-P22065

On Classiﬁcation of Fermionic

Rational Conformal Field Theories

Zhihao Duan, Kimyeong Lee, Sungjay Lee and Linfeng Li

Korea Institute for Advanced Study

85 Hoegiro, Dongdaemun-Gu, Seoul 02455, Korea

Abstract

We systematically study how the integrality of the conformal characters

shapes the space of fermionic rational conformal ﬁeld theories in two dimen-

sions. The integrality suggests that conformal characters on torus with a given

choice of spin structures should be invariant under a principal congruence sub-

group of PSL(2,Z). The invariance strongly constrains the possible values of

the central charge as well as the conformal weights in both Neveu-Schwarz and

Ramond sectors, which improves the conventional holomorphic modular boot-

strap method in a signiﬁcant manner. This allows us to make much progress on

the classiﬁcation of fermionic rational conformal ﬁeld theories with the number

of independent characters less than ﬁve.

arXiv:2210.06805v2 [hep-th] 8 Jun 2023

Contents

1 Introduction and Conclusion 1

2 Preliminaries 4

2.1 Modular linear diﬀerential equations . . . . . . . . . . . . . . . . . . 8

2.2 Integrality conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Candidate conformal weights . . . . . . . . . . . . . . . . . . . . . . . 15

3 Classiﬁcation 19

3.1 Rank 2 with ℓ≤1 ............................ 21

3.2 Rank 3 with ℓ≤1 ............................ 23

3.3 Rank 4 with ℓ=0 ............................ 25

A Some Finite Group Theory 28

B Exponents 29

1 Introduction and Conclusion

Conformal ﬁeld theories (CFTs) play prominent roles in theoretical physics ranging

from the critical phenomena of phase transitions, the boundary excitation of (frac-

tional) quantum Hall eﬀects, to the world-sheet dynamics of quantum strings. If we

specialize to two dimensions, the conformal symmetry gets enlarged to the inﬁnite

dimensional Virasoro algebra that restricts the underlying dynamics severely. As a

consequence, for example, any unitary CFT with central charge cless than one can

only be one of the minimal models [1].

However, for general 2d CFTs, it still remains a diﬃcult task to comprehend the

full landscape of their theory space. After the breakthrough in the study of 3d Ising

model [2], the philosophy of bootstrap has taken a more central role in recent years.

In particular, it became more compelling to explore the idea of modular bootstrap,

which employs the modular invariance of the CFT partition functions on the torus.

The modular invariance implies new constraints on the space of 2d CFTs as well as

possible 3d gravity duals, and also provides unexpected connections to mathematics.

See for example [3–15] for a very partial list of related works.

Rational conformal ﬁeld theories (RCFTs), deﬁned to have ﬁnitely many chiral

primaries, have drawn particular attention in the past decades. The minimal models,

lattice CFTs, and the Wess-Zumino-Witten (WZW) models of compact group are

just a few examples of the RCFT. Compared to irrational CFTs, they have much

nicer properties, and we are naturally led to a quest for its possible classiﬁcation.

Mathur, Mukhi and Sen [16] ﬁrst realized that modular symmetry can be explored

to systematically classify bosonic RCFTs, based on the number of independant char-

acters d, the rank of RCFT, they have. This approach is dubbed as holomorphic

modular bootstrap [17–19]. Its essential idea is the observation that characters form

a vector-valued modular function (vvmf), and mathematically they must satisfy a

modular linear diﬀerential equation (MLDE). We also remark that the method of

MLDE also features in the study of higher dimensional quantum ﬁeld theories, most

notably in four-dimensional superconformal ﬁeld theories (SCFTs) through the so-

called SCFT/VOA correspondence [20–25].

On the other hand, in mathematics there is a class of diﬀerent although related

objects: modular tensor categories (MTCs). RCFTs and MTCs are similar because

modularity plays important role in both cases. For instance the modular data of an

MTC should in some sense capture the modular transformation of RCFT characters.

The classiﬁcation of MTCs according to their rank is also an important topic [26–28],

which resonates with the holomorphic modular bootstrap method. However, since it

is still unknown whether every MTC is realized by an RCFT, and even if so a given

MTC could be mapped to many RCFTs, we are still lacking a precise dictionary

between the two methods.

Nevertheless, this does not stop us from utilizing techniques developed in the MTC

side. In particular, the so-called congruence property [29] is utilized to constrain

the set of modular data for MTC at low ranks. The latest result can be found

in [30]. In the RCFT side, this can be formulated as the integrality conjecture or

unbounded denominator conjecture [31], which was recently proved in [32] (see also

[33]). Its statement is that each component of a vvmf becomes a modular function

for a congruence subgroup Γ(N) of SL(2,Z) if all the coeﬃcients in its q-expansion

are integral. Last year, [34] imported this technique to RCFTs and greatly extended

the previous classiﬁcation using holomorphic modular bootstrap.

In this paper, we will consider a generalization of the above successful story to

theories including fermions. This generalization was initiated in the papers [35,36], in

which one examines the modular subgroups preserved by the choice of spin structure

for fermions, and write down corresponding fermionic MLDEs (FMLDEs). Naturally,

this tool helps to classify what one may call fermionic RCFTs (FRCFTs), which has

interesting connections to various topics such as fermionization [37,38], emergence of

supersymmetry (SUSY) [39–42], moonshine phenomena of sporadic groups [36, 43],

etc.

As a next step, naturally we would like to study the implication of integrality

of the Fourier coeﬃcients of the characters for FRCFTs. We state an analogue of

the integrality conjecture in 2.2, which is the counterpart of congruence property in

super-MTC [44]. Assuming this conjecture, we are able to, in certain sense, extend the

previous classiﬁcation in [35, 36], and we successfully bootstrap candidate solutions

for putative FRCFT characters up to rank four. To be more speciﬁc, ﬁrst of all, by

our working hypothesis, we are able to cover non-degenerate FRCFTs, meaning that

there exists no pair of NS sector conformal weights whose diﬀerence is a multiple of a

half-integer, and no pair of R sector conformal weights whose diﬀerence is an integer.

For this class, then indeed all the theories found in [35,36] are recovered. Moreover,

there is a non-negative integer ℓas another input parameter, which characterizes the

pole structure of the coeﬃcient functions of FMLDE and is known as the index. We

only consider index ℓ≤1 in this paper, together with possible unitarity constraint.

For easy of reference, we summarize the main results here:

(d, ℓ) = (2,0) (d, ℓ) = (3,0) unitary (d, ℓ) = (3,1) unitary (d, ℓ) = (4,0) unitary

Tables 3, 4 Table 5 Table 6 Table 7

We would like to stress here that the previous approaches to the classiﬁcation of

the FRCFTs as well as some studies in the super-MTC are limited in a sense that

they mainly rely on the physical constraints in the Neveu-Schwarz (NS) sector but

barely concern those in the Ramond (R) sector. There are however some occasions

where the torus partition function in the NS sector, that looks perfectly consistent,

is modular-transformed to the partition function in the R sector that is ill-deﬁned.

Recently, it was pointed out in [45] that careful examining the often-ignored Ramond

sector results in a stronger constraint on the spectra of fermionic CFTs. Interestingly,

we observe that the consequence of the integrality conjecture for FRCFTs actually

not only constrains the spectra in both NS and R sectors but also the provides a

consistency relation between them.

For readers who wish to compare our results with the classiﬁcation in the super-

MTC literature, we ﬁrst remark that the exponential of conformal weights in their

language are known as twists or topological spins, while their central charge cis only

deﬁned mod 8. Also, to obtain their normalization of Tmatrix in the modular data,

one needs to multiply ours by an overall factor exp(c/24). Therefore, the number N

which labels the congruence subgroup will in general diﬀer from those appearing in

the modular data. More importantly, due to the fact that diﬀerent primaries may

share the same character, the rank of super-MTC will in general be bigger than the

number of independent characters in our classiﬁcation.

As possible further directions, ﬁrst it would be nice to understand or disprove

the solutions that we ﬁnd but are unable to identify. For instance, one has to check

if they have the well-deﬁned fusion algebra. Although each fusion coeﬃcient can

be computed by the Verlinde formula, it requires the so-called reﬁned modular ma-

trices. However, the conformal characters constructed by the holomorphic modular

bootstrap only provide the reduced modular matrices that leads to the wrong fusion

coeﬃcients. See [46] for reference. It would be interesting to develop a systematic

manner to unfurl the reduced modular matrices to reﬁned modular matrices that even-

tually furnishes the (super) MTC data, whose classiﬁcation can be found in [47,48].

Second, one could consider turning on extra parameters such as ﬂavor fugacity in

the characters, which upgrades the MLDEs to the so-called ﬂavored MLDEs. Such

generalization has already appeared, for example, in [22, 24]. Third, it would be in-

teresting to look for a correct Hecke operator relating diﬀerent FRCTs, as was done

for bosonic RCFTs in [49, 50]. Finally, incorporating all possible topological defect

lines for fermionic theories would be another important direction to pursue [51], and

FRCFTs are surely suitable examples to study.

This paper is organized as follows. In section 2, we review the basic structure of 2d

CFT in the presence of fermions, and introduce the holomorphic modular bootstrap

method. New ingredients start from section 2.2 where we make use of the integral-

ity conjecture and hence the representation theory of ﬁnite groups to constrain all

possible exponents mod 1 for fermionic MLDEs at low ranks. Section 3 contains

the main result of this paper, where we present explicitly putative two-, three- and

four-character fermionic theories with constraint mentioned above after a computer-

ized scan. In particular, they contain all non-degenerate theories previously found

in [35, 36]. We also include two appendices. In Appendix A, we give some detail

about the induced representation to prove a claim in Section 2.2. In Appendix B, we

list all the exponents mod 1 that are used in Section 3.

Note Added: While this work is at the ﬁnal stage, [52] appeared which uses the

same idea to construct modular data in the super-MTC.

2 Preliminaries

An RCFT is deﬁned as a CFT whose torus partition function can be described as

a ﬁnite sum of products of holomorphic functions and anti-holomorphic functions of

the complex parameter of torus τ,

Z(τ, ¯τ) = TrHqL0−c/24 ¯q¯

L0−c/24i=

d−1

i,j=0

Mijχi(τ)¯χj(¯τ) (1)

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KIAS-P22065OnClassificationofFermionicRationalConformalFieldTheoriesZhihaoDuan,KimyeongLee,SungjayLeeandLinfengLiKoreaInstituteforAdvancedStudy85Hoegiro,Dongdaemun-Gu,Seoul02455,KoreaAbstractWesystematicallystudyhowtheintegralityoftheconformalcharactersshapesthespaceoffermionicrationalconformalfield...

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KIAS-P22065 On Classification of Fermionic Rational Conformal Field Theories

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