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A Class of Topological Actions

M. Bauer, G. Girardi, R. Stora, F. Thuillier

TL;DR

This work develops a comprehensive framework to describe topological actions in gauge theories using Cheeger-Simons differential characters and Deligne-Beilinson cohomology. It shows how circulation of $U(1)$ fields and higher $p$-form potentials can be encoded as Deligne-Beilinson cohomology classes, and introduces Defining and Long formulas for integrating these classes over cycles or the whole manifold, including smoothing via partitions of unity. The formalism is extended to distributional coefficients, enabling rigorous semiclassical and quantum-field treatments. Together, these results clarify a Pontryagin-type duality between cycles and differential characters and provide a robust mathematical foundation for topological actions in physics.

Abstract

We review definitions of generalized parallel transports in terms of Cheeger-Simons differential characters. Integration formulae are given in terms of Deligne-Beilinson cohomology classes. These representations of parallel transport can be extended to situations involving distributions as is appropriate in the context of quantized fields.

A Class of Topological Actions

TL;DR

This work develops a comprehensive framework to describe topological actions in gauge theories using Cheeger-Simons differential characters and Deligne-Beilinson cohomology. It shows how circulation of fields and higher -form potentials can be encoded as Deligne-Beilinson cohomology classes, and introduces Defining and Long formulas for integrating these classes over cycles or the whole manifold, including smoothing via partitions of unity. The formalism is extended to distributional coefficients, enabling rigorous semiclassical and quantum-field treatments. Together, these results clarify a Pontryagin-type duality between cycles and differential characters and provide a robust mathematical foundation for topological actions in physics.

Abstract

We review definitions of generalized parallel transports in terms of Cheeger-Simons differential characters. Integration formulae are given in terms of Deligne-Beilinson cohomology classes. These representations of parallel transport can be extended to situations involving distributions as is appropriate in the context of quantized fields.

Paper Structure

This paper contains 25 sections, 147 equations.

Table of Contents

  1. Introduction
  2. Maxwell Semi-Classical theory (à la Feynman)
  3. Integral representations of differential characters
  4. Circulation of $U(1)$ gauge fields as differential characters
  5. Integration over a cycle: the appearance of Deligne-Beilinson classes
  6. Examples
  7. An equivalent integration over the whole manifold
  8. Smoothing
  9. Integration of Deligne-Beilinson classes with distributional coefficients
  10. Conclusions
  11. Deligne-Beilinson cohomology
  12. Definitions and notations
  13. Deligne-Beilinson cochains
  14. Computation of $H^{q}(\mathcal{C}_p,D)$, $q < p$
  15. The $\check{\hbox{C}}$ech homotopy operator
  16. ...and 10 more sections