Toward the Pole

TL;DR

This paper explores the analytic structure of semiclassical conformal blocks and their holographic interpretation via geodesic configurations in AdS3, revealing non-analytic behaviors and singularity resolutions through resummation techniques.

Contribution

It provides a novel analysis of conformal block discontinuities and their holographic interpretation, connecting resummation of instanton contributions to geodesic networks in AdS3.

Findings

Discontinuities in conformal blocks are linked to geodesic configurations.

Resummation removes singularities in the conformal blocks.

Order parameter exhibits non-analytic behavior near poles.

Abstract

We study the analytic structure of semiclassical conformal blocks, namely of the 1-point conformal block on the torus and of the 4-point conformal block on the sphere, as functions of the intermediate dimension. We interpret their discontinuities, which can be revealed with the use of a particular resummation procedure, holographically in terms of configurations of geodesics in AdS3. In other words, we study the behavior of the $\Omega$-deformed $\mathcal N=2$ SYM theory in the Nekrasov-Shatashvili limit near those singular points, where naively the W-boson with non-vanishing angular momentum becomes massless. Upon a proper resummation of instanton contributions, these singularities disappear, which is similar to the Seiberg-Witten solution in the undeformed case where there is no massless W-boson. It is shown that the order parameter undergoes a non-analytic behavior near positions of the poles. The jump of the order parameter is interpreted holographically in terms of geodesic networks