Relative Topological Integrals and Relative Cheeger-Simons Differential Characters

TL;DR

This paper extends the framework of Cheeger--Simons differential characters to manifolds with boundary, enabling the proper treatment of topological integrals in field theories involving boundaries.

Contribution

It introduces a computable construction of relative Cheeger--Simons differential characters that generalizes the classical absolute case.

Findings

Provides a new mathematical framework for topological integrals on manifolds with boundary.

Contains the classical Cheeger--Simons differential characters as a special case.

Facilitates applications in string and D-brane theories with boundary conditions.

Abstract

Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of (absolute) (co)homology using the formalism of Cheeger--Simons differential characters. String and D--brane theory involve field theoretic models on worldvolumes with boundary. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger--Simons differential characters. In this paper, we present a construction of relative Cheeger--Simons differential characters which is computable in principle and which contains the ordinary Cheeger--Simons differential characters as a particular case.