Virasoro blocks and quasimodular forms

TL;DR

This paper demonstrates that Virasoro conformal blocks in the heavy intermediate exchange limit can be expressed as polynomials in Eisenstein series, revealing deep connections with modular forms and enabling efficient recursive computations.

Contribution

It introduces a closed-form polynomial representation of Virasoro blocks in the heavy limit using Eisenstein series, linked to modular anomaly equations and the 2d/4d correspondence.

Findings

Virasoro blocks are expressible as Eisenstein series polynomials

Modular anomaly equations explain the polynomial structure

New recursive algorithm for constructing blocks in the heavy limit

Abstract

We analyse Virasoro conformal blocks in the regime of heavy intermediate exchange $(h_p \rightarrow \infty)$. For the 1-point block on the torus and the 4-point block on the sphere, we show that each order in the large-$h_p$ expansion can be written in closed form as polynomials in the Eisenstein series. The appearance of this structure is explained using the fusion kernel and, more markedly, by invoking the modular anomaly equations via the 2d/4d correspondence. We observe that the existence of these constraints allows us to develop a faster algorithm to recursively construct the blocks in this regime. We then apply our results to find corrections to averaged heavy-heavy-light OPE coefficients.