Mass formula for topological boundary conditions from TQFT gravity

TL;DR

This paper introduces a mass formula for topological boundary conditions in 3d TQFTs, linking it to the TQFT partition function and extending to non-Abelian and higher-dimensional theories.

Contribution

It generalizes the concept of mass formulas to topological boundary conditions in TQFTs, providing explicit calculations for Abelian, non-Abelian, and 5D theories.

Findings

Mass formula reduces to code mass formulas in Abelian cases.

Explicit evaluation of mass for Abelian theories and Ising modular tensor category.

Extension of the mass concept to five-dimensional theories.

Abstract

Mass formulas evaluate the total weighted count of a given class of algebraic structures, such as lattices or codes. We show that 3d TQFTs provide a generalization of this concept: the total weighted count of topological boundary conditions is given by the TQFT partition function averaged over all closed 3d manifolds. This weighted count, which we call the mass, can be interpreted as the renormalized partition function of TQFT gravity. For Abelian TQFTs, the mass formula for topological boundary conditions reduces to the mass formula for particular families of codes. Focusing on the Abelian case, we show how to evaluate the mass for any bosonic theory and consider many explicit examples. We then discuss the non-Abelian generalization and compute the mass for $n + \bar n$ copies of the Ising modular tensor category. Finally, we generalize the construction to five dimensions and compute the mass for Abelian 2-form Chern-Simons theories.