2010/04/30 by Valentin Bonzom, Matteo Smerlak
Mathematics · Physics and Astronomy · #Abelian group #Betti number #Black Holes and Theoretical Physics #Cohomology #Degree (music) #Dynamical systems theory #Haar measure #Homotopy and Cohomology in Algebraic Topology #Measure (data warehouse) #Noncommutative and Quantum Gravity Theories #Path integral formulation #Quantum field theory #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1007/s11005-010-0414-4
published as Lett.Math.Phys.93:295-305,2010 · 5 pages
openalex publication_date 2010/07/28 · arxiv created 2010/09/16 · arxiv updated 2014/11/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory, the Ponzano-Regge model of 3d quantum gravity and discrete BF theory, 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 in general, because of a phenomenon called `bubble divergences'. A common expectation is that the degree of these divergences is given by the number of `bubbles' of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian -- in both cases, the divergence degree is given by the second Betti number of the 2-complex.