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Discrete Deligne Cohomology and Discretized Abelian Chern-Simons Theory


Norton, Thomas Clark (2022) Discrete Deligne Cohomology and Discretized Abelian Chern-Simons Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/aae8-re02.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Discretization; Triangulation; Deligne Cohomology; Differential Cohomology; Cheeger-Simons Characters; Line Bundles; Discrete Line Bundles
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marcolli, Matilde
Thesis Committee:
  • Ni, Yi (chair)
  • Marcolli, Matilde
  • Gukov, Sergei
  • Pei, Du
Defense Date:26 August 2021
Record Number:CaltechTHESIS:09072021-163418163
Persistent URL:
Norton, Thomas Clark0000-0003-1438-8408
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14356
Deposited By: Thomas Norton
Deposited On:15 Oct 2021 16:07
Last Modified:05 Jul 2022 19:09

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