Modular properties of characters of the W3 algebra

TL;DR

The paper establishes exact modular transformation laws for traces of powers of the W3 zero mode, including $\mathrm{Tr}(W_0)$, $\mathrm{Tr}(W_0^2)$, and $\mathrm{Tr}(W_0^3)$, across Verma and minimal-model representations. It derives modular differential equations for traces with a single $W_0$ insertion and shows that normalized traces are weight-0 modular forms with nonnegative $q$-expansions, while the unnormalised traces form weight-3 modular objects. By combining Gaberdiel–Hartman–Jin methods with Virasoro-primary techniques, the authors obtain exact transformation laws that relate $W_0^2$ and $W_0^3$ transforms to derivatives acting on Virasoro characters, and verify consistency with GHJ results. The large-central-charge analysis reveals two consistent large-$c$ regimes, corresponding to $h_{ m min}=0$ or $h_{ m min}=c/24$, realized respectively by real-coupling Toda theory and by non-unitary minimal models, with implications for holography and black-hole microstate studies in higher-spin AdS$_3$ gravity.

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.