On simplicial \(BF\) theory
This paper is a short summary of the author's constructions and results in [``Notes on simplicial \(BF\) theory'', \url{arXiv:hep-th/0610326v3}], which were later extended in his Ph. D. thesis [``Discrete \(BF\) theory'', \url{arXiv:0809.1160v2}], and may serve as an introduction or as a guide to these works. The author's results fit into the framework of A. Losev's simplicial programme for topological quantum field theory, whose goal is to replace the Lagrangian formalism of field theory, in particular of Chern-Simons theory, by a simplicial version in which the space of fields is finite-dimensional. This would allow us to calculate the partition function and correlation functions of quantum field theory using finite-dimensional integrals rather than functional ones, avoiding divergence and renormalization problems. The author's work concerns a simplified toy model for topological field theory: non-abelian BF theory without observables. The Batalin-Vilkovisky formalism involves replacing the forms in the classical BF action by non-homogeneous differential forms. The induced action for simplicial BF theory is shown to split into the sum of local contributions of the individual simplices of the triangulation of our manifold \(M\). Therefore, to know the action, it suffices to perform one universal computation for one simplex in each dimension. This computation is carried out in dimensions \(0\) and \(1\), and perturbative results are obtained for higher dimensions.
scientific article; zbMATH DE number 5485369