Exact WKB Analysis of ${\mathbb{C}\mathbb{P}}^1$ Holomorphic Blocks

We study holomorphic blocks in the three dimensional $\mathcal{N}=2$ gauge theory that describes the ${\mathbb{C}\mathbb{P}}^1$ 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 ${}_1{\varphi }_1$ $q$-hypergeometric function. We show in detail how the $q$-Borel resummation methods reproduce the results obtained previously by using block-integral methods.