Abstract
We revisit certain natural algebraic transformations on the space of 3D topological quantum field theories (TQFTs) called “Galois conjugations.” Using a notion of multiboundary entanglement entropy (MEE) defined for TQFTs on compact 3-manifolds with disjoint boundaries, we give these abstract transformations additional physical meaning. In the process, we prove a theorem on the invariance of MEE along orbits of the Galois action in the case of arbitrary Abelian theories defined on any link complement in S3. We then give a generalization to non-Abelian TQFTs living on certain infinite classes of torus link complements. Along the way, we find an interplay between the modular data of non-Abelian TQFTs, the topology of the ambient spacetime, and the Galois action. These results are suggestive of a deeper connection between entanglement and fusion.
Highlights
Suited and interesting arena in which to apply this approach
In addition to preserving fusion rules of topological quantum field theories (TQFTs), Galois conjugation preserves multiboundary entanglement entropy (MEE) in broad classes of theories
We showed that putting any Abelian TQFT on any link complement in S3 and tracing out Hilbert spaces on any subset of the links leads to an invariant MEE along Galois orbits
Summary
As alluded to in the introduction, large classes of 3D TQFTs have a description in terms of algebraic objects called MTCs. The importance of this algebraic description of a TQFT is that it allows us to think of a TQFT as a solution to a finite number of polynomial equations rather than being tied to a Lagrangian. MTC exists only if a set of consistency conditions called the Pentagon and Hexagon equations are satisfied [2,3,4,5]. These relations arise due to the commutativity and associativity of the fusion operation. The solutions to these polynomials belong to a finite extension of the rational numbers, K [14] This property enables us to use Galois theory to define a map, called Galois conjugation, from one TQFT to another
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