From torus bundles to particle-hole equivariantization

We continue the program of constructing (pre)modular tensor categories from 3manifolds first initiated by Cho-Gang-Kim using M theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved in the construction is a collection of certain SL(2, C) characters on a given manifold, which serve as the simple object types in the corresponding category. Chern-Simons invariants and adjoint Reidemeister torsions also play a key role, 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 Z(2)-equivariantization of certain pointed premodular categories. Here the equivariantization is performed for the 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.

Automated summary

This paper continues the program of constructing (pre)modular tensor categories from 3-manifolds using M theory and mathematical methods, and discusses the important structures involved and the challenges faced. By considering a specific class of 3-manifolds, the paper demonstrates how to realize the modular data using equivariantization.