On simplicial \(BF\) theory
type: Item
instance of: scholarly article
title: On simplicial \(BF\) theory
zbMATH Open document ID: 1209.57024
DOI: 10.1134/S1064562408010158
author: P. N. Mnev
published in: Doklady Mathematics
publication date: 2009-01-05T00:00:00Z
review text: 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.
reviewed by: Daniel Moskovich
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 81T45
Mathematics Subject Classification ID: 81T70
zbMATH DE Number: 5485369
zbMATH Keywords: BF theory
zbMATH Keywords: TQFT
zbMATH Keywords: Batalin-Vilkovisky
zbMATH Keywords: quantization
MaRDI profile type: Publication
cites work: The geometry of the master equation and topological quantum field theory.
cites work: Topological BF theories in 3 and 4 dimensions
cites work: Simplicial De Rham cohomology and characteristic classes of flat bundles
cites work: Geometry of Batalin-Vilkovisky quantization
full work available at URL: s1064562408010158
OpenAlex ID: W2026865801
description: scientific article; zbMATH DE number 5485369