Abstract
In this work we revisit the “holomorphic modular bootstrap”, i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2, ℤn), we describe a method to classify allowed central charges and weights (c, hi) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, hi). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.
Highlights
In the general program of classifying rational conformal field theories (RCFTs) [1], substantial effort has been put into classifying the following proxies: modular tensor categories (MTCs) [2, 3] and vector-valued modular functions [4].1 The two are intimately related: the modular S- and T-matrices of an modular tensor category (MTC) capture, in some appropriate sense, the PSL(2, Z) transformation properties of RCFT characters, which form a vvmf.2 Each proxy has its limitations
In this work we revisit the “holomorphic modular bootstrap”, i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters
By making use of the representation theory of PSL(2, Zn), we describe a method to classify allowed central charges and weights (c, hi) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, hi)
Summary
In the general program of classifying rational conformal field theories (RCFTs) [1], substantial effort has been put into classifying the following proxies: modular tensor categories (MTCs) [2, 3] and vector-valued modular functions (vvmfs) [4].1 The two are intimately related: the modular S- and T-matrices of an MTC capture, in some appropriate sense, the PSL(2, Z) transformation properties of RCFT characters, which form a vvmf. Each proxy has its limitations. As will be explained below, we will be able to restrict to meromorphic functions φk(τ ) with certain types of singularities, in which case the space of such modular forms becomes finite-dimensional This allows one to classify vvmfs by constructing and solving the most general MDE of a given order, e.g. as a Fourier series in q := e2πiτ. For non-rigid MDEs, PSL(2, Zn) representation theory still permits a classification of allowed exponents for integral characters, and we carry this classification out In this case imposing physical constraints such as positivity and existence of a vacuum becomes more difficult.
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