Discrete Connection Laplacian Energy

• Discrete Connection Laplacian Energy is a spectral invariant that quantifies field interactions on discrete structures such as graphs, simplicial complexes, and tori.
• It computes energy by summing over eigenvalues derived from local parallel transport data, linking geometry with combinatorial methods.
• Its analysis connects theta functions, zeta functions, and determinant regularizations, bridging discrete models and continuum spectral geometry.

The discrete connection Laplacian energy is a spectral invariant associated with discrete analogues of connection Laplacians on graphs, simplicial complexes, or periodic structures such as discrete tori. This energy quantitatively characterizes field interactions and geometric structure by associating a sum or trace over the eigenvalues of a connection Laplacian—an operator defined using local parallel transport data—on sections of vector bundles (or more generally, modules over division algebras) over discrete combinatorial spaces. Fundamental connections exist between the Laplacian energy, spectral zeta functions, theta traces, and determinant regularizations, and its analysis underpins advances in spectral geometry, random geometric representations, and manifold learning.

1. Discrete Connection Laplacians: Definitions and Structure

The discrete connection Laplacian is an operator acting on sections of a vector bundle (or a division-algebra module) over a finite combinatorial structure such as a simplicial complex or a discrete torus. Given a bundle with a unitary (or normed-invertible) flat connection, parallel transport maps are associated with adjacent vertices or higher-dimensional faces.

For a finite discrete torus $DT_N = \mathbb{Z}^n / M_N \mathbb{Z}^n$ (with $M_N$ an invertible integer $n \times n$ matrix), and a flat $O(d)$-connection, the connection Laplacian $\Delta^{\mathrm{conn}}_N$ acts as

$$(\Delta^{\mathrm{conn}}_N V)([x]) = \sum_{y \sim x} (V([x]) - \phi_{xy} V([y]))$$

where $[x]$ ranges over the torus, $y \sim x$ denotes adjacency, $V([x])$ is a bundle section, and $\phi_{xy}$ is the orthogonal parallel transport between fibers.

Analogously, for an abstract simplicial complex $M_N$0 and division algebra $M_N$1, with a field $M_N$2 (the unit sphere), the connection Laplacian $M_N$3 and its adjoint $M_N$4 are defined entrywise by

$M_N$5

where $M_N$6, $M_N$7 is the core, and $M_N$8 is the star of $M_N$9 (Knill, 2020, Lin et al., 2024).

2. Discrete Connection Laplacian Energy: Formulation and Properties

The discrete connection Laplacian energy is typically defined as the sum of the eigenvalues (trace) of the connection Laplacian operator over the combinatorial space:

$n \times n$0

For the discrete torus, in terms of the explicit spectral decomposition, this yields

$n \times n$1

where $n \times n$2 are torsion shifts determined by the holonomy, and $n \times n$3 is the dual lattice (Lin et al., 2024).

In the setting of division algebra-valued fields on simplicial complexes, the total energy is

$n \times n$4

A central result shows that this total energy is precisely the sum of the field values, $n \times n$5, regardless of the algebraic structure of $n \times n$6 (Knill, 2020).

For point-cloud constructions sampled from a manifold, the discrete connection Laplacian energy (in Dirichlet/graph form) for a sampled frame field $n \times n$7 is

$n \times n$8

where $n \times n$9 are kernel-based edge weights and $O(d)$0 are discrete parallel transport maps (Singer et al., 2013).

3. Spectral Theory and Asymptotic Behavior

Eigenvalue analysis of the discrete connection Laplacian reveals that its spectrum organizes according to combinatorial and connection data. For discrete tori, all eigenvalues are real, non-negative, and explicit sinusoidal formulas are available:

$O(d)$1

with multiplicity determined by holonomy shifts.

In the large-torus limit ($O(d)$2) with appropriate rescalings, eigenvalues $O(d)$3 converge pointwise to those of the real-torus connection Laplacian. The per-vertex energy converges to a geometric constant:

$O(d)$4

where $O(d)$5 is the torus dimension, $O(d)$6 the fiber dimension (Lin et al., 2024).

In the vector bundle and simplicial complex setting, over $O(d)$7, the set of Laplacian matrices with simple spectrum forms a non-compact Kähler manifold of dimension $O(d)$8, with spectral permutations corresponding to fundamental group symmetries (Knill, 2020).

4. Connections to Theta Functions, Zeta Functions, and Determinants

The Laplacian energy is tightly linked to spectral zeta and theta functions. For finite discrete structures, the theta trace is defined as

$O(d)$9

The energy is the negative rate of decay at zero:

$\Delta^{\mathrm{conn}}_N$0

The spectral zeta function,

$\Delta^{\mathrm{conn}}_N$1

satisfies $\Delta^{\mathrm{conn}}_N$2, relating the energy to analytic continuations of the spectrum. The regularized determinant is given by

$\Delta^{\mathrm{conn}}_N$3

Asymptotics for the regularized determinant express leading-order growth in terms of integrals of Bessel functions, the dimension of the kernel, and the continuum analog (Lin et al., 2024).

5. Determinant Structure and Noncommutative Generalizations

In contexts where the Laplacian matrix takes values in noncommutative division algebras, the ordinary determinant does not apply. The Dieudonné determinant provides an alternative, mapping from $\Delta^{\mathrm{conn}}_N$4 matrices over a division ring $\Delta^{\mathrm{conn}}_N$5 to the abelianization $\Delta^{\mathrm{conn}}_N$6. For connection Laplacians $\Delta^{\mathrm{conn}}_N$7 constructed from a field $\Delta^{\mathrm{conn}}_N$8,

$\Delta^{\mathrm{conn}}_N$9

This result ensures a unimodularity-type property connecting combinatorial, connection, and spectral structure, with Laplace-type expansion and Cauchy-Binet compatibility in the noncommutative case (Knill, 2020).

6. Discrete-to-Continuum Convergence and Principal Bundle Extensions

For discrete samples from principal bundle structures over manifolds, discrete connection Laplacians and their energies converge (in probability) to continuum counterparts as the number of samples increases and the connection structure is suitably approximated. Explicitly,

$$(\Delta^{\mathrm{conn}}_N V)([x]) = \sum_{y \sim x} (V([x]) - \phi_{xy} V([y]))$$0

with bias and variance decaying with sample complexity and bandwidth parameter, ensuring that both spectral and energy properties of the discrete operator asymptotically reflect smooth manifold geometry (Singer et al., 2013).

Generalizations include arbitrary principal $$(\Delta^{\mathrm{conn}}_N V)([x]) = \sum_{y \sim x} (V([x]) - \phi_{xy} V([y]))$$1-bundles with orthogonal holonomy representations, maintaining the convergence framework and extending applicability to a wide range of geometric and algebraic contexts.

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References:

(Knill, 2020) "Division algebra valued energized simplicial complexes" (Singer et al., 2013) "Spectral Convergence of the connection Laplacian from random samples" (Lin et al., 2024) "Connection Laplacian on discrete tori with converging property"