Affine SL(2) conformal blocks from 4d gauge theories

TL;DR

This work shows that including a full surface operator in 4d N=2 SU(2) quiver gauge theories converts Nekrasov's instanton partition function into a modified affine SL(2) conformal block. In the Nekrasov–Shatashvili limit (ε2→0), these affine blocks furnish simultaneous eigenfunctions of the quantized Hitchin Hamiltonians, linking gauge theory, affine CFT, and integrable systems. Explicit SU(2) examples demonstrate the appearance of an insertion K and connect torus one-point blocks to the elliptic Calogero-Moser system, suggesting a broad framework for understanding surface operators through conformal blocks and Hitchin quantization. The results bridge the AGT-like correspondence with Hitchin-system quantization, offering a path to generalize to higher N and to explore related S^4/Pestun-like structures.

Abstract

We study Nekrasov's instanton partition function of four-dimensional N=2 gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) Lie algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.

Figures (3)

• Figure 1: The setup of Nekrasov-Shatashvili and the addition of a surface operator. We insert the surface operator on the plane $z_2=0$, which is shown as an unshaded parallelogram. In the limit $\epsilon_2\to 0$, we have an effectively two-dimensional system on the plane $z_1=0$, shown as an shaded parallelogram.
• Figure 2: Pictorial representation of the propagator $K^{-1}$, shown in (a), the vertex with two external legs $R$, shown in (b) and the vertex with one external leg, shown in (c). External legs are represented with solid lines, while internal legs are represented with dashed lines.
• Figure 3: Sewing of building blocks into the four point conformal block on the sphere