Towards a classification of two-character rational conformal field theories

Published for SISSA by Springer Received: November 6, 2018 Revised: January 29, 2019 Accepted: April 19, 2019 Published: April 29, 2019 A. Ramesh Chandra and Sunil Mukhi Indian Institute of Science Education and Research, Homi Bhabha Rd, Pashan, Pune 411 008, India E-mail: , Abstract: We provide a simple and complete construction of infinite families of consistent, modular-covariant pairs of characters satisfying the basic requirements to describe twocharacter RCFT. These correspond to solutions of generic second-order modular linear differential equations. To find these solutions, we first construct “quasi-characters” from the Kaneko-Zagier equation and subsequent works by Kaneko and collaborators, together with coset dual generalisations that we provide in this paper. We relate our construction to the Hecke images recently discussed by Harvey and Wu. Keywords: Conformal and W Symmetry, Conformal Field Theory, Field Theories in Lower Dimensions ArXiv ePrint: 1810.09472 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2019)153 JHEP04(2019)153 Towards a classification of two-character rational conformal field theories Contents 1 2 One-character CFT 2.1 Basic features 2.2 Modular linear differential equations 4 4 5 3 Two characters: complete classification for ` < 6 3.1 The general theory and the map to hypergeometric equations 3.2 The case of ` = 1, 3, 5 3.3 The case of ` = 0 3.4 The case of ` = 2 3.5 The case of ` = 4 3.6 Fusion rules 3.7 Comparison to previous work 8 8 11 11 16 18 19 21 4 Quasi-characters 4.1 How characters fail 4.2 Kaneko-Zagier parametrisation 4.3 Lee-Yang series and its dual 4.4 A1 series and its dual 4.5 A2 series and its dual 4.6 D4 series and its dual 4.7 General approach to recursion relations 4.8 Modular transformations for quasi-characters 4.9 Coset relations among quasi-characters 4.10 Summary of the properties of quasi-characters 22 22 24 26 28 30 30 31 33 34 35 5 Generating characters for ` ≥ 6 5.1 Multiplicative method 5.2 Additive method 5.3 Completeness of the additive method 5.4 Relation to Hecke images 36 36 37 40 42 6 Conclusions and discussion 44 A Rademacher series for vector-valued modular functions 45 B Examples of quasi-characters B.1 Lee-Yang class, ` = 0 B.2 Dual Lee-Yang class: ` = 2 B.3 A1 class, ` = 0 46 46 48 50 –i– JHEP04(2019)153 1 Introduction B.4 B.5 B.6 B.7 B.8 1 A1 class, ` = 2 A2 class, ` = 0 A2 class, ` = 2 D4 class, ` = 0 D4 class, ` = 2 51 53 54 55 57 Introduction 1 Our use of “admissible” may differ from some of the mathematical literature. –1– JHEP04(2019)153 The traditional classification of Rational Conformal Field Theories (RCFT) in 2d is based on either (chiral) symmetry algebras or lattices. In the former approach, after selecting a particular chiral algebra or set of algebras, one uses the structure of null vectors to write down a minimal series of CFT’s that furnish realisations (not necessarily unitary) of the given algebra [1, 2] (for more examples see [3] and references therein). Among possible chiral algebras, Kac-Moody algebras are special because one can consider affine models containing all integrable primaries at a given level, namely WZW models, and then take cosets of them. This coset construction generates a vast supply of RCFT’s including the various minimal series already obtained by null-vector methods. The approach based on lattices includes the famous c = 1 classification [4, 5] and the constructions based on even, self-dual lattices as reviewed for example in [6]. These two approaches, the chiral algebra/WZW coset approach and the lattice approach, are not complete — we know RCFT’s that belong in neither of these classes [7, 8]. Moreover, there are simple coset models which do not come from affine WZW theories [9]. Hence it can be useful to organise RCFT’s not by these criteria but by simplicity, defined as the number of independent scaling exponents or primary fields, roughly the same as the number of characters. The c = 24 models classified in [10] are examples of one-character CFT’s where there is an extremely large chiral algebra, including many spin-2 generators. The only primary field of this extended algebra is the identity. Several two-character RCFT’s have likewise long been known [11, 12]. A complete classification of the simplest models could be more useful than the traditional approach for various purposes, including applications in condensed-matter systems as well as for studies from a mathematical viewpoint. An additional application of simple rational CFT’s has emerged in the context of four-dimensional supersymmetric gauge theories [13–15]. An approach to the classification of RCFT with a small numbers of characters was proposed long ago in [11]. It is based on the observation that the characters of a rational CFT are vector-valued modular forms of weight zero and satisfy a Modular Linear Differential Equation (MLDE) (this fact was independently noted in [16]). Thus, to classify them we may start from a general class of MLDE and identify values of the parameters for which they determine “admissible” sets of candidate characters — those whose q-expansion has non-negative integer coefficients.1 After finding admissible characters there are well-defined –2– JHEP04(2019)153 procedures, using the modular transformation matrix S and its relation to the fusion rules via the Verlinde formula [17], to determine whether they truly describe a consistent RCFT, and if so to solve the resulting theories. These procedures were implemented in considerable detail in [12, 18] and some recent work in this direction can be found in [19]. There is also now a considerable mathematical literature on using MLDE to find vector-valued modular forms of CFT type, of which some relevant works are [20–28] and additional ones will be described in what follows. Modular linear differential equations are characterised by an integer n, the number of characters, which is also the order of the differential equation, as well as an integer ` ≥ 0, ` 6= 1 which specifies the number of zeroes in moduli space of the Wronskian determinant of the solutions (this concept will be explained in more detail below). In [11] all admissible characters with n = 2, ` = 0 were classified and form a finite set. All but one have been identified with definite RCFT’s, while one has a degenerate vacuum and was initially rejected for this reason. It is now understood to be an Intermediate Vertex Operator Algebra (IVOA) [29] based on the notion [30] of Intermediate Lie Algebra. Related mathematical work on the 2nd order ` = 0 equation may be found in [31–36]. Subsequently the case n = 2, ` = 2 was classified in [12, 37]. Again there is a finite set of admissible characters. A remarkable property of these characters, shown in [9], is that they are in one to one correspondence with the ` = 0 theories and satisfy a “generalised coset” relationship with them (...truncated)