Mass formula for topological boundary conditions from TQFT gravity

TL;DR

The paper defines a mass for 3d TQFTs as a weighted count of topological boundary conditions and shows that, in the Abelian case, this mass equals a renormalized TQFT gravity partition function summed over all closed 3-manifolds. It develops a practical framework to compute this mass by summing over Heegaard splittings, exploiting modular invariance and Gauss-sum techniques, and demonstrates that the result reduces to classical mass formulas for self-dual codes in many Abelian theories. The authors extend the construction to non-Abelian TQFTs, illustrating the Ising case, and generalize the approach to five dimensions with Abelian 2-form theories, connecting boundary data to stabilizer codes and symplectic codes. Overall, the mass serves as a bridge between topological field theory, coding theory, and holographic ensembles, offering explicit combinatorial and algebraic tools to classify boundary data and suggesting further extensions to richer theories and higher dimensions.

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.