The Liouville side of the Vortex

TL;DR

The paper establishes a precise link between non-abelian vortex counting in four-dimensional $\mathcal{N}=2$ quiver gauge theories and Liouville/Toda conformal blocks with degenerate insertions via the AGT correspondence and open topological string theory. It shows that the full vector space of degenerate conformal blocks is reproduced by the four-dimensional limit of open string strip amplitudes with quiver boundary conditions, and identifies the non-abelian vortex partition function with a specific fusion channel of those blocks. The analysis extends from SU(2) to SU($N$), deriving corresponding vortex and simple surface operator correspondences and connecting Toda fusion rules to quiver data. The results provide a unified CFT/topological string/gauge theory framework for vortex counting and surface operators, with potential extensions to refined strings and integrable systems. The approach offers a pathway to interpret conformal blocks as 4D gauge theory data and to explore defects via topological string boundary conditions.

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.

Figures (11)

• Figure 1: AGT relation between $SU(2)$ quiver gauge theory and CFT
• Figure 2: $SU(2)$ vortex from quiver theory
• Figure 3: $SU(2)$ simple surface operators from quiver theory
• Figure 4: relation between $SU(2)$ vortex and CFT
• Figure 5: relation between $SU(2)$ simple surface operator and CFT