Spin Modular Categories

TL;DR

This paper introduces new structures on modular categories that enable refined quantum 3-manifold invariants incorporating cohomology and spin structures, generalizing existing invariants and providing a splitting formula.

Contribution

It develops a framework for defining refined quantum invariants using invertible objects in modular categories, unifying known cohomological and spin refinements.

Findings

All known refinements are special cases of the new construction

A splitting formula for refined invariants is established

The approach generalizes product decompositions of quantum invariants

Abstract

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is played by objects which are invertible under tensor product. All known examples of cohomological or spin type refinements of the Witten-Reshetikhin-Turaev 3-manifold invariants are special cases of our construction. In addition, we establish a splitting formula for the refined invariants, generalizing the well-known product decomposition of quantum invariants into projective ones and those determined by the linking matrix.