Remarks about Connection and Dirac matrices

TL;DR

The paper develops a unified linear-algebraic framework for finite simplicial complexes by studying the Dirac matrix $D$, the connection matrix $L$, and their Laplacians, together with dynamical extensions $D_T$, $L_T$, and $g_T$ under simplicial maps. It establishes unimodularity of $L$, explicit Green-function inverses, and interlacing/spectral bounds, then connects these operators to topological invariants such as the Euler characteristic and Lefschetz fixed-point theory, including a discrete McKean–Singer symmetry. It further explores wave equations, discrete time evolutions, and dynamical zeta-type quantities, and extends the discussion to higher (quadratic) cohomology with corresponding Dirac operators and conjectures about weak Loewner dominance among $L$, $D$, and $g$. The work outlines several open questions, including weak-Loewner dominance implications, uniqueness of maximal eigenvalues of $g=L^{-1}$, and potential derivations of fixed-point results from quadratic cohomology, with practical relevance to discrete dynamical systems on complexes. The overall contribution lies in linking spectral graph theory, topological invariants, and discrete dynamics within a coherent, unimodular matrix framework.

Abstract

The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the eigenvalues for subcomplexes. This gives general upper bounds of the eigenvalues both for L and D in terms of inclusion or intersection degrees. We conjecture that L always dominates both D and the inverse of L in a weak Loewner sense. In a second part we look at dynamical systems (G,T), where T is a simplicial map on G. Both L and D generalize to dynamical versions of L and D. The modified L is still unimodular with an explicit Green function inverse and modified Dirac part still comes from an exterior derivative d. We also review the Lefschetz fixed point theorem for a simplicial map T on a simplicial complex G which implies the Brouwer fixed point theorem: any simplicial map on a contractible finite abstract simplicial complex G has a fixed simplex.

Figures (2)

• Figure 1: The figure show sthe spectra $\lambda$ (left) and cumulative spectra $S$ (right) of the connection matrix $L$, the Green matrix $g=L^{-1}$ and the Dirac matrix $D=d+d^*$ for a random complex $G$. $L$ appears to weakly Loewner dominate both $D$ and $g$.
• Figure 2: The estimates $\lambda_j(D) \leq d_j(D)$ and $\lambda_j(L) \leq d_j(L)$ where $d_j(D)$ and $d_j(L)$ are the ordered vertex degrees of $D$ and $L$.

Theorems & Definitions (37)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 1
• proof