Remarks about Connection and Dirac matrices

TL;DR

This paper explores the spectral properties of connection Laplacians and Dirac matrices derived from finite simplicial complexes, establishing eigenvalue bounds, conjecturing dominance relations, and extending concepts to dynamical systems with fixed point implications.

Contribution

It introduces new eigenvalue bounds for connection Laplacians and Dirac matrices, proposes a dominance conjecture, and extends these operators to dynamical systems with fixed point theorems.

Findings

Eigenvalue interlacing for subcomplexes

Upper bounds for eigenvalues based on degrees

Fixed point theorems for simplicial maps

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.