A new construction of $c=1$ Virasoro blocks

TL;DR

This paper presents a novel method connecting $c=1$ Virasoro blocks to Heisenberg blocks via a nonabelianization map using spectral networks, leading to new formulas and eigenblock constructions.

Contribution

It introduces a nonabelianization map relating $c=1$ Virasoro blocks to Heisenberg blocks on branched covers, providing new formulas and eigenblock constructions.

Findings

New formulas for Virasoro blocks in terms of fermion correlation functions.

Intertwining of the nonabelianization map with Verlinde loop operators.

Construction of eigenblocks and new $ au$-function formulas.

Abstract

We introduce a nonabelianization map for conformal blocks, which relates $c=1$ Virasoro blocks on a Riemann surface $C$ to Heisenberg blocks on a branched double cover $\widetilde{C}$ of $C$. The nonabelianization map uses the datum of a spectral network on $C$. It gives new formulas for Virasoro blocks in terms of fermion correlation functions determined by the Heisenberg block. The nonabelianization map also intertwines with the action of Verlinde loop operators, and can be used to construct eigenblocks. This leads to new Kyiv-type formulas and regularized Fredholm determinant formulas for $\tau$-functions.