Galois Orbits of TQFTs: Symmetries and Unitarity

TL;DR

This paper explores how Galois actions influence 2+1D topological quantum field theories, focusing on unitarity preservation, Galois orbits, and the construction of fixed point theories through symmetry gauging.

Contribution

It characterizes Galois actions on TQFTs, establishes conditions for unitarity preservation, and links fixed point theories to gauging symmetries in abelian TQFTs.

Findings

Galois actions can preserve unitarity under certain conditions.

Identification of trivial Galois orbits as fixed point TQFTs.

All fixed point TQFTs can be constructed via symmetry gauging of abelian theories.

Abstract

We study Galois actions on $2+1$D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and 2-groups. In order to gain a better physical understanding of Galois actions, we prove sufficient conditions for the preservation of unitarity. We then map out the Galois orbits of various classes of unitary TQFTs. The simplest such orbits are trivial (e.g., as in various theories of physical interest like the Toric Code, Double Semion, and 3-Fermion Model), and we refer to such theories as unitary "Galois fixed point TQFTs." Starting from these fixed point theories, we study conditions for preservation of Galois invariance under gauging 0-form and 1-form symmetries (as well as under more general anyon condensation). Assuming a conjecture in the literature, we prove that all unitary Galois fixed point TQFTs can be engineered by gauging 0-form symmetries of theories built from Deligne products of certain abelian TQFTs.