Discrete BF theory

TL;DR

This paper develops a discrete version of non-abelian BF theory on triangulated manifolds, enabling finite-dimensional calculations of topological quantities and relating the theory to $qL_$ structures on cell cochains.

Contribution

It introduces a finite-dimensional discrete BF theory framework for triangulated manifolds, connecting it to $qL_$ algebraic structures and simplifying calculations of topological invariants.

Findings

Discrete BF theory allows finite-dimensional integrals for topological quantities.

Effective action on cohomology can be computed explicitly.

Discrete BF action relates to $qL_$ structures via homotopy transfer.

Abstract

In this work we discuss the simplicial program for topological field theories for the case of non-abelian BF theory. Discrete BF theory with finite-dimensional space of fields is constructed for a triangulated manifold (or for a manifold equipped with cubical cell decomposition), that is in a sense equivalent to the topological BF theory on manifold. This discrete version allows one to calculate interesting quantities from the BF theory, like the effective action on cohomology, in terms of finite-dimensional integrals instead of functional integrals, as demonstrated in a series of explicit examples. We also discuss the interpretation of discrete BF action as the generating function for $qL_\infty$ structure (certain "one-loop version" of ordinary $L_\infty$ algebra) on the cell cochains of triangulation, related to the de Rham algebra of the underlying manifold by homotopy transfer procedure. This work is a refinement of older text hep-th/0610326.