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Simplicial Cheeger-Simons models and simplicial higher abelian gauge theory

Jyh-Haur Teh

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.

Simplicial Cheeger-Simons models and simplicial higher abelian gauge theory

Abstract

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

Paper Structure

This paper contains 24 sections, 16 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Cheeger--Simons models and spark groups
  3. Cheeger--Simons models
  4. General spark complexes
  5. Discrete Cheeger--Simons differential characters
  6. Whitney forms and induced cochain maps
  7. Cheeger--Simons triangulations
  8. Comparison maps and convergence
  9. Discrete higher abelian gauge theory
  10. Combinatorial Hodge theory on cochains
  11. Determinants and theta functions
  12. Partition functions
  13. Inverse limit identification
  14. Worked examples and computations
  15. Flat rectangular tori $T^n$: harmonic lattice and theta functions
  16. ...and 9 more sections

Key Result

Lemma 2.5

Let $G$ be a divisible abelian group. If $g\in \mathop{\mathrm{Hom}}\nolimits(I_k,G)$ satisfies $\delta g=0$ and $g|_{Z_k(I_{\bullet})}=0$, then $g=\delta g'$ for some $g'\in \mathop{\mathrm{Hom}}\nolimits(I_{k-1},G)$.

Theorems & Definitions (45)

  • Definition 2.1: Cheeger--Simons model
  • Example 2.2
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 35 more