Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model

TL;DR

This paper extends string functions in logarithmic conformal field theory, introducing higher functions related to W-algebra characters, and analyzes their modular properties and decomposition of characters.

Contribution

It introduces higher string functions A_{n,r} and B_{n,r} for the ^sl(2)_k/u(1) coset, generalizing classic string functions and exploring their modular transformations.

Findings

Higher string functions generalize classic string functions.

Decomposition involves non-holomorphic, non-quasiperiodic functions.

Modular group representation exhibits features of higher-level Appell functions.

Abstract

We generalize the string functions C_{n,r}(tau) associated with the coset ^sl(2)_k/u(1) to higher string functions A_{n,r}(tau) and B_{n,r}(tau) associated with the coset W(k)/u(1) of the W-algebra of the logarithmically extended ^sl(2)_k conformal field model with positive integer k. The higher string functions occur in decomposing W(k) characters with respect to level-k theta and Appell functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-functions). The decomposition coefficients, to be considered ``logarithmic parafermionic characters,'' are given by A_{n,r}(tau), B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra characters of the (p=k+2,1) logarithmic model. We study the properties of A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string functions C_{n,r}, and evaluate the modular group representation generated from A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular transformations of the higher-level Appell functions and the associated transcendental function Phi.