From Torus Bundles to Particle-Hole Equivariantization

• URL: http://arxiv.org/abs/2106.01959v3
• Date: Tue, 27 Sep 2022 15:14:06 GMT
• Title: From Torus Bundles to Particle-Hole Equivariantization
• Authors: Shawn X. Cui, Paul Gustafson, Yang Qiu, Qing Zhang
• Abstract summary: We consider an infinite family of 3-manifolds, that is, torus bundles over the circle. We show that the modular data are realized by the $mathbbZ$-equivariantization of certain pointed premodular categories. It is our hope that this extensive class of examples will shed light on how to improve the program to recover the full data of a premodular category.
• Score: 15.857538570676667
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We continue the program of constructing (pre)modular tensor categories from 3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved is a collection of certain $\text{SL}(2, \mathbb{C})$ characters on a given manifold which serve as the simple object types in the corresponding category. Chern-Simons invariants and adjoint Reidemeister torsions play a key role in the construction, and they are related to topological twists and quantum dimensions, respectively, of simple objects. The modular $S$-matrix is computed from local operators and follows a trial-and-error procedure. It is currently unknown how to produce data beyond the modular $S$- and $T$-matrices. There are also a number of subtleties in the construction which remain to be solved. In this paper, we consider an infinite family of 3-manifolds, that is, torus bundles over the circle. We show that the modular data produced by such manifolds are realized by the $\mathbb{Z}_2$-equivariantization of certain pointed premodular categories. Here the equivariantization is performed for the $\mathbb{Z}_2$-action sending a simple (invertible) object to its inverse, also called the particle-hole symmetry. It is our hope that this extensive class of examples will shed light on how to improve the program to recover the full data of a premodular category.

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