Non-invertible twisted compactification of class $\mathcal S$ theory and $(B,B,B)$ branes

TL;DR

This paper explores a novel non-invertible twisted compactification of class S theories, revealing that the resulting 3d N=4 sigma models have target spaces as special branes within Hitchin moduli space, with explicit examples provided.

Contribution

It introduces a new non-invertible symmetry defect in class S theories and characterizes the resulting branes as fixed points in Hitchin moduli space, with concrete calculations for specific cases.

Findings

The 3d theory is a sigma model with a hyperKähler target space.

The target space is a $(B,B,B)$ brane, a fixed point set of a finite subgroup of the mapping class group.

Explicit examples for type A1, genus 2 class S theory are computed.

Abstract

We study non-invertible twisted compactification of class $\mathcal S$ theories on $S^1$: we insert a non-invertible symmetry defect at $S^1$ extending along remaining directions and then compactify on $S^1$. We show that the resulting 3d theory is 3d $\mathcal N=4$ sigma model whose target space is a hyperK\"ahler submanifold of Hitchin moduli space, i.e. a $(B,B,B)$ brane. The $(B,B,B)$ brane is the fixed point set on Hitchin moduli space of a finite subgroup of mapping class group of underlying Riemann surface. We describe the $(B,B,B)$ branes as affine varieties and calculate concrete examples of these $(B,B,B)$ branes for type $A_1$, genus $2$ class $\mathcal S$ theory.