Stokes Phenomena in 3d $\mathcal{N}=2$ SQED$_2$ and $\mathbb{CP}^1$ Models

TL;DR

This paper investigates the Stokes phenomena in the holomorphic blocks of the $ ext{CP}^1$ model by analyzing it as a decoupling limit of SQED$_2$, revealing symmetry transformations and block relations that elucidate the Stokes structure.

Contribution

It introduces a novel approach using $ ext{Z}_3$ symmetry to relate holomorphic blocks of SQED$_2$ and $ ext{CP}^1$ models, uncovering their Stokes phenomena and block transformations.

Findings

Identified six pairs of SQED$_2$ holomorphic blocks and their Stokes-like regions.

Derived the relation between SQED$_2$ blocks and $ ext{CP}^1$ blocks in a decoupling limit.

Reproduced the Stokes regions and matrices of $ ext{CP}^1$ blocks from SQED$_2$ analysis.

Abstract

We propose a novel approach of uncovering Stokes phenomenon exhibited by the holomorphic blocks of $\mathbb{CP}^1$ model by considering it as a specific decoupling limit of SQED$_2$ model. This approach involves using a $\mathbb{Z}_3$ symmetry that leaves the supersymmetric parameter space of SQED$_2$ model invariant to transform a pair of SQED$_2$ holomorphic blocks to get two new pairs of blocks. The original pair obtained by solving the line operator identities of the SQED$_2$ model and the two new transformed pairs turn out to be related by Stokes-like matrices. These three pairs of holomorphic blocks can be reduced to the known triplet of $\mathbb{CP}^1$ blocks in a particular decoupling limit where two of the chiral multiplets in the SQED$_2$ model are made infinitely massive. This reduction then correctly reproduces the Stokes regions and matrices of the $\mathbb{CP}^1$ blocks. Along the way, we find six pairs of SQED$_2$ holomorphic blocks in total, which lead to six Stokes-like regions covering uniquely the full parameter space of the SQED$_2$ model.