Bubble Divergences from Cellular Cohomology |
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Authors: | Valentin Bonzom and Matteo Smerlak |
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Institution: | (1) Department of Mathematics, University of California, Berkeley, CA, 94720-3840, U.S.A;(2) KdV Institute for Mathematics, University of Amsterdam, Science Park, 904 1098 Amsterdam, XH, The Netherlands;(3) Department of Mathematics, University of Aarhus, Aarhus, Denmark |
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Abstract: | 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. |
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