Discrete BF theory

Understand

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.