Modular properties of characters of the W3 algebra

TL;DR

This paper investigates the modular properties of characters in the W3 algebra, providing exact formulas for their transformations and differential equations, revealing connections to weight 0 modular forms with non-negative integer coefficients.

Contribution

It derives exact modular transformation formulas for traces of powers of W0 in the W3 algebra and relates differential equations of these traces to known modular forms.

Findings

Exact modular transformation formulas for W0 traces

Differential equations for traces with W0 insertions

Connections to weight 0 modular forms with non-negative coefficients

Abstract

In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of $W_0^n$ for n = 1, 2, 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single $W_0$ inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.