The Liouville side of the Vortex

TL;DR

This paper connects Liouville conformal blocks with vortex partition functions, showing their equivalence via topological string amplitudes and revealing new insights into the structure of conformal field theories and gauge theories.

Contribution

It demonstrates that conformal blocks with degenerate insertions are fully reproduced by topological string amplitudes and identifies vortex partition functions with specific conformal block channels.

Findings

Conformal blocks are represented by topological string amplitudes.

Vortex partition functions correspond to fusion channels of conformal blocks.

The approach bridges conformal field theory and gauge theory via string theory.

Abstract

We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensional limit of open topological string amplitudes on the strip with generic boundary conditions associated to a suitable quiver gauge theory. As a byproduct we identify the non-abelian vortex partition function with a specific fusion channel of degenerate conformal blocks.