Classifying three-character RCFTs with Wronskian index equalling 0 or 2

Abstract

In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers [n,l]. n is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and l, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study [3,0] and [3,2] MLDEs in order to classify the corresponding CFTs. We reduce the problem to a “finite” problem: to classify CFTs with central charge 0 < c ≤ 96, we need to perform 6, 720 computations for the former and 20, 160 for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity.In the [3,0] case, for 0 < c ≤ 96, we obtain many character-like solutions: two infinite classes and a discrete set of 303. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify [3,2] CFTs for 0 < c ≤ 96: each CFT in this case is obtained by adjoining a constant character to a [2,0] CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.

Bootstrapping fermionic rational CFTs with three characters

“…Note added. While this work is completed, [27,28] appeared which contain some overlap in the classification of bosonic RCFT with three characters. In this paper, we use the analytic expressions of the solutions to establish the classification.…”

Needles in a haystack: An algorithmic approach to the classification of 4d $\mathcal{N}=2$ SCFTs

“…In the context of 2d CFT, this idea goes back to the classic work of Mathur, Mukhi, and Sen [45][46][47], see e.g. [43,44,[48][49][50][51][52][53][54][55][56][57][58][59][60][61] for recent developments. As scanning over the set of real numbers parameterizing the MDE would clearly be impossible, our first step is to map the problem to a scan over integers.…”

New meromorphic CFTs from cosets

“…From the study of MLDE it emerged that an important classifier for RCFT with a given number of characters is an integer ≥ 0, = 1 called the Wronskian index (for a detailed review, see [3]). Admissible characters for bosonic CFTs have been constructed in [4][5][6][7][8][9][10][11][12] and for fermionic CFTs in [13][14][15].…”

“…Let us briefly review some basic aspects of the meromorphic coset relation (more details can be found in [10,19]). The numerator theory H is typically an extension of a non-simple Kac-Moody algebra ⊕ i G r i ,k i by higher-spin generators that organise a subset of Kac-Moody characters into a single character.…”

“…Next, we remove C from the story and find bilinear relations where C is paired with the characters of another known CFT C . For n = 3, large numbers of such coset pairs were generated in [10]. Since C is a different theory from C (and not a tensor product of C with something else), the result is a new meromorphic theory at a different central charge.…”