Remarks about Connection and Dirac matrices

2026-01-26

AI Summary

This study investigates the spectral properties of the connection Laplacian \(L\) and the Dirac operator \(D\) on finite simplicial complexes, along with eigenvalue interlacing phenomena between subcomplexes, and extends these analyses to a dynamical systems framework defined by simplicial maps. By integrating tools from algebraic topology, spectral graph theory, and matrix analysis—particularly through explicit constructions of Green’s functions and applications of the Lefschetz fixed-point theorem—we establish universal upper bounds for the eigenvalues of both \(L\) and \(D\). We further formulate and partially verify a conjecture asserting that \(L\) dominates \(D\) and \(L^{-1}\) in the weak Loewner order. Key contributions include a generalized formulation preserving the unimodularity of \(L\) and the exterior differential structure of \(D\) within dynamical systems, leading to a discrete analogue of Brouwer’s fixed-point theorem.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• connection Laplacian
• Dirac matrix
• eigenvalue interlacing
• simplicial complex
• Lefschetz fixed point theorem

Innovation

• connection Laplacian
• Dirac matrix
• eigenvalue interlacing
• dynamical simplicial complex
• Lefschetz fixed point theorem