Simplicial Cheeger-Simons models and simplicial higher abelian gauge theory

A pair $(K,K^{\mathrm{\prime }})$ consisting of a smooth triangulation $K$ of a compact smooth oriented Riemannian manifold $M$ and a sufficiently fine subdivision $K^{\mathrm{\prime }}$ determines a finite-dimensional Cheeger–Simons model $\mathcal{C}\mathcal{S}(K,K^{\mathrm{\prime }})$ built from Whitney-type data on the induced curvilinear complexes. Its associated differential character groups ${\text{Diff}}^{\bullet }(\mathcal{C}\mathcal{S}(K,K^{\mathrm{\prime }}))$ 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 $K^{\mathrm{\prime }}$ for which $(K,K^{\mathrm{\prime }})$ 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 ${\text{Diff}}^k(\mathcal{C}\mathcal{S}(K,K^{\mathrm{\prime }}))$ approximate the identity in a Sobolev-dual seminorm as $\text{mesh}(K^{\mathrm{\prime }})\to 0$. For closed $M$, we further identify ${\widehat{H}}^k(M)$ canonically with the inverse limit of ${\text{Diff}}^k(\mathcal{C}\mathcal{S}(K,K^{\mathrm{\prime }}))$ over refinements. As an application, we formulate a simplicial higher abelian gauge theory whose gauge-invariant configuration space is ${\text{Diff}}^p(\mathcal{C}\mathcal{S}(K,K^{\mathrm{\prime }}))$, and we prove that the resulting simplicial (regularized) partition function converges, in the refining limit, to the corresponding smooth regularized partition function of Kelnhofer.