Abstract
We study Galois actions on 2+1D 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.
Highlights
Symmetries provide a non-perturbative way to constrain the dynamics of a quantum field theory (QFT)
In a general topological quantum field theories (TQFTs) described by a modular tensor category C, a gapped boundary corresponds to the condensation of a subset of anyons in C which admits the structure of a Lagrangian algebra [64]
We explored several aspects of Galois actions on TQFTs and gave a sufficient condition for producing unitary Galois orbits
Summary
Symmetries provide a non-perturbative way to constrain the dynamics of a quantum field theory (QFT). Finding a consistent 2 + 1D TQFT in this sense essentially follows from finding the zeros of some multivariable polynomials Though these constraints are often too complicated to be solved exactly, a general mathematical result (the Ocneanu rigidity theorem) states that there are only a finite number of inequivalent solutions to the Pentagon and Hexagon equations for a given set of fusion rules [10]. We look at how Galois conjugation interacts with gauging 0-form symmetries and anyon condensation We use these results to characterize Galois invariant TQFTs. Section 5 contains several additional examples of Galois conjugation of TQFTs which concretely illustrate our ideas and compliment our discussion. There is partial overlap of this work with our section 3
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