Web of dualities on non-orientable surfaces

TL;DR

This paper explores the web of dualities in two-dimensional bosonic theories with non-anomalous $ ext{Z}_2$ and time-reversal symmetries, extending fermionization to non-orientable surfaces and analyzing the resulting structures.

Contribution

It proves that the group of topological manipulations forms a dihedral group $D_8$ and systematically investigates the duality web via Symmetry TFT and sector actions.

Findings

Fermionization extends to non-orientable surfaces with $ ext{Pin}^-$ structures.

The group of topological manipulations is isomorphic to $D_8$.

Dualities are analyzed through Symmetry TFT and $S^1$ Hilbert space actions.

Abstract

It is known that a two-dimensional bosonic theory with a non-anomalous $\mathbb{Z}_2$ symmetry can be fermionized. Recent work shows that if the bosonic theory also has non-anomalous time-reversal symmetry, fermionization extends to non-orientable surfaces and yields a fermionic theory that depends on a $\mathrm{Pin}^-$ structure. Besides fermionization, one can define various topological manipulations, such as gauging and stacking invertible phases, which together generate a web of dualities. We prove that their group structure is the dihedral group $D_8$ of order 16. Furthermore, we systematically investigate the web from two perspectives: Symmetry TFT and actions on sectors of the $S^1$ Hilbert space.