Simplicial Cheeger-Simons models and simplicial higher abelian gauge theory

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.

💡 Research Summary

The paper develops a rigorous finite‑dimensional approximation of Cheeger–Simons differential characters on a compact oriented Riemannian manifold (M). Starting from a smooth triangulation (K) of (M) and a sufficiently fine subdivision (K’), the authors construct a Cheeger–Simons model (\mathscr{CS}(K,K’)=(E_\bullet(L,L’),I_\bullet(L’),\mathcal Z)) using Whitney forms on the curvilinear complexes (L) and (L’) induced by the triangulation. The cochain complex (E_\bullet(L,L’)) is the image of the Whitney map from the original simplicial chains, while (I_\bullet(L’)) consists of integral singular chains on the refined complex. The pairing (\mathcal Z) is given by integration of Whitney forms over integral chains.

They prove that (\mathscr{CS}(K,K’)) satisfies the axioms of a Cheeger–Simons model: the chain complex is free, the pairing is non‑degenerate, there are no non‑integral periods, Stokes’ formula holds, and the de Rham isomorphism between cohomology of (E_\bullet) and that of (\operatorname{Hom}(I_\bullet,\mathbb R)) is an isomorphism. Consequently one can define differential character groups (\Diff^{k}(\mathscr{CS}(K,K’))) exactly as in the classical theory.

Two natural maps are introduced: a discretization map (\iota:\widehat H^{k}(M)\to\Diff^{k}(\mathscr{CS}(K,K’))) obtained by pulling back a smooth differential character to Whitney cochains, and an extension map (\epsilon:\Diff^{k}(\mathscr{CS}(K,K’))\to\widehat H^{k}(M)) obtained by solving a de Rham problem to lift a discrete curvature form to a smooth form. Under a uniform shape‑regularity (fullness) condition on the triangulations, the authors show that as the mesh size (\operatorname{mesh}(K’)\to0) the compositions (\epsilon\circ\iota) and (\iota\circ\epsilon) converge to the identity in a Sobolev‑dual seminorm (H^{-s}) for suitable (s>0). This quantitative result establishes that the finite‑dimensional model approximates the smooth theory arbitrarily well.

For closed manifolds, they consider the directed system of all Cheeger–Simons triangulations ((K,K’)) and prove that the inverse limit (\varprojlim_{(K,K’)}\Diff^{k}(\mathscr{CS}(K,K’))) is canonically isomorphic to the original differential character group (\widehat H^{k}(M)). The proof relies on the stability of cohomology under refinement and the fact that the comparison maps are compatible with the direct system.

The paper then applies this machinery to higher abelian gauge theory. For a fixed degree (p), the gauge‑invariant configuration space is taken to be (\Diff^{p}(\mathscr{CS}(K,K’))). The action functional is a discretized version of Kelnhofer’s regularized higher‑form gauge action, expressed in terms of the curvature representative in (E_{p+1}(L,L’)) and the integral class in (H_{p+1}(I_\bullet(L’))). Observables are defined analogously to the smooth case, depending only on the differential character class. The authors prove that the simplicial (regularized) partition function (Z_{K,K’}) converges, as the mesh goes to zero, to the smooth regularized partition function (Z_{\text{cont}}) of Kelnhofer. The convergence proof uses the Sobolev‑dual approximation of curvature forms and the continuity of the determinant‑torsion‑theta function factors that appear in the regularized functional integral.

Finally, explicit examples on the torus and the sphere are worked out, illustrating how the discrete theta‑functions and torsion factors reproduce the known smooth results. Numerical experiments confirm the theoretical convergence rates.

In summary, the authors provide a complete axiomatic framework that bridges smooth Cheeger–Simons differential characters and finite‑dimensional simplicial models, prove precise approximation theorems, identify the inverse limit with the original smooth group, and demonstrate the utility of this construction by building a convergent simplicial higher‑form abelian gauge theory. This work offers both a mathematically rigorous discretization scheme for differential cohomology and a concrete path toward regularized functional integrals in higher‑form gauge theories.