From torus bundles to particle–hole equivariantization

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 in the construction is a collection of certain SL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {SL}(2, \mathbb {C})$$\end{document} 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 Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document}-equivariantization of certain pointed premodular categories. Here the equivariantization is performed for the Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document}-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.