Simplicial Cheeger-Simons models and simplicial higher abelian gauge theory

Abstract

A pair $(K,K')$ consisting of a smooth triangulation $K$ of a compact smooth oriented Riemannian manifold $M$ and a sufficiently fine subdivision $K'$ determines a finite-dimensional Cheeger--Simons model $\mathscr{CS}(K,K')$ built from Whitney-type data on the induced curvilinear complexes. Its associated differential character groups $\Diff^{\bullet}(\mathscr{CS}(K,K'))$ provide a simplicial, finite-dimensional counterpart of the Cheeger--Simons differential characters $\widehat H^{\bullet}(M)$. We prove that every smooth triangulation admits a subdivision $K'$ for which $(K,K')$ is a Cheeger--Simons triangulation in this sense. Under a uniform fullness (shape-regularity) hypothesis, we show that the natural discretization/extension maps between $\widehat H^{k}(M)$ and $\Diff^{k}(\mathscr{CS}(K,K'))$ approximate the identity in a Sobolev-dual seminorm as $\mesh(K')\to 0$. For closed $M$, we further identify $\widehat H^{k}(M)$ canonically with the inverse limit of $\Diff^{k}(\mathscr{CS}(K,K'))$ over refinements. As an application, we formulate a simplicial higher abelian gauge theory whose gauge-invariant configuration space is $\Diff^{p}(\mathscr{CS}(K,K'))$, and we prove that the resulting simplicial (regularized) partition function converges, in the refining limit, to the corresponding smooth regularized partition function of Kelnhofer.