# Exact WKB Analysis of $\mathbb{CP}^1$ Holomorphic Blocks

## TL;DR

This paper applies exact WKB analysis to study holomorphic blocks in a 3D ${
m N}=2$ gauge theory for the $ ext{CP}^1$ model, deriving analytic continuation formulas and connecting them with $q$-hypergeometric functions.

## Contribution

It introduces a novel application of exact WKB methods to analyze holomorphic blocks and their analytic continuations in the $ ext{CP}^1$ gauge theory context.

## Key findings

- Derived analytic continuation formulas for holomorphic blocks.
- Connected $q$-Borel resummation with previous block-integral results.
- Utilized connection formulas for ${}_1 ext{phi}_1$ $q$-hypergeometric functions.

## Abstract

We study holomorphic blocks in the three dimensional ${\mathcal N}=2$ gauge theory that describes the $\mathbb{CP}^1$ model. We apply exact WKB methods to analyze the line operator identities associated to the holomorphic blocks and derive the analytic continuation formulae of the blocks as the twisted mass and FI parameter are varied. The main technical result we utilize is the connection formula for the ${}_1\phi_1$ $q$-hypergeometric function. We show in detail how the $q$-Borel resummation methods reproduce the results obtained previously by using block-integral methods.