Fuzzy Geometry via the Spinor Bundle, with Applications to Holographic Space-time and Matrix Theory

TL;DR

This paper introduces a novel fuzzy geometric framework based on Dirac operators, enabling rotation-invariant approximations without symplectic forms, with applications to holographic space-time and matrix models.

Contribution

It generalizes fuzzy geometry using Dirac and Dirac-Flux operators, applicable to arbitrary spheres and motivated by holographic principles.

Findings

Framework is rotation invariant on n-spheres

Does not require symplectic forms

Applicable to holographic space-time and matrix theory

Abstract

We present a new framework for defining fuzzy approximations to geometry in terms of a cutoff on the spectrum of the Dirac operator, and a generalization of it that we call the Dirac-Flux operator. This framework does not require a symplectic form on the manifold, and is completely rotation invariant on an arbitrary n-sphere. The framework is motivated by the formalism of Holographic Space-Time (HST), whose fundamental variables are sections of the spinor bundle over a compact Euclidean manifold. The strong holographic principle (SHP) requires the space of these sections to be finite dimensional. We discuss applications of fuzzy spinor geometry to HST and to Matrix Theory.