Universal Modular Properties of Generalized Gibbs Ensembles and Chiral Deformations

Abstract

We study modular properties of conformal field theories perturbed by holomorphic fields. We prove an asymptotic formula for the modular S-transform of a generalized partition function that includes zero modes of higher spin holomorphic currents. The derivation makes use of general properties of torus correlation functions, in particular the Zhu recursion relation. The asymptotic expansion of the modular transformed partition function takes a universal form that is determined iteratively by the second order pole coefficients in the operator product expansion of the holomorphic currents. This proves and generalizes a conjecture regarding the modular transformation properties of generalized Gibbs ensembles.

Figures (1)

• Figure 1: (a) A torus with $A$ and $B$ cycles realized as a quotient of the plane in two ways (b) and (d), with $u' = u\tau + 1 \equiv u\tau$, and correspondingly as an annulus in two ways with (c) $z=\exp(2\pi i u)$, and (e) $z' = \exp(2\pi i u')$. The initial integration over $u$ along the (red) spatial A cycle from 0 to 1 in (b) as in equation \ref{['Aperiodintegrals']} becomes an integration from 0 to $\tau$ in (d) and over $z'$ from 1 to $q=\exp(2\pi i \tau)$ in (e) as in equation \ref{['Bperiodintergrals']}.