Discrete Deligne Cohomology and Discretized Abelian Chern-Simons Theory

Author: Norton, Thomas Clark

Year: 2022

Degree: Dissertation (Ph.D.)

Advisor: Marcolli, Matilde

Committee Members: Ni, Yi; Marcolli, Matilde; Gukov, Sergei; Pei, Du

Option: Mathematics

DOI: 10.7907/aae8-re02

Abstract

The differential cohomology groups of a smooth manifold are discretized with respect to a triangulation. The realization of differential cohomology used is Deligne cohomology. A discretized version of the smooth Deligne double complex is constructed from cochain groups defined on simplices of the triangulation. The total cohomology of this double complex is studied and shown to satisfy exact sequences analogous to the standard structural sequences satisfied by differential cohomology. In the degree corresponding to line bundles with connection, our cohomology classes are shown to correspond to isomorphism classes of an existing notion of discrete line bundles with connection. Explicit examples of these discrete line bundles with connection are constructed. A ring structure is defined on the discrete Deligne cohomology groups; it is graded-commutative and non-associative (however, associativity is recovered in the continuum limit). The ring structure allows one to define a more general discrete Chern-Simons action than has previously appeared in the literature.

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