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Bubble divergences from twisted cohomology. (English) Zbl 1257.83011
Summary: We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory and 3d Riemannian quantum gravity, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined because of a phenomenon known as ‘bubble divergences’. In this paper, we extend recent results of the authors to the cases where these divergences cannot be understood in terms of cellular cohomology. We introduce in its place the relevant twisted cohomology, and use it to compute the divergence degree of the partition function. We also relate its dominant part to the Reidemeister torsion of the complex, thereby generalizing previous results of J. W. Barrett and I. Naish-Guzman [Classical Quantum Gravity 26, No. 15, Article ID 155014, 48 p. (2009; Zbl 1172.83017)]. The main limitation to our approach is the presence of singularities in the representation variety of the fundamental group of the complex; we illustrate this issue in the well-known case of two-dimensional manifolds.

MSC:
83C45 Quantization of the gravitational field
83C75 Space-time singularities, cosmic censorship, etc.
83C27 Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
81T20 Quantum field theory on curved space or space-time backgrounds
83C80 Analogues of general relativity in lower dimensions
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
81T13 Yang-Mills and other gauge theories in quantum field theory
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
81S40 Path integrals in quantum mechanics
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