Topology at the Planck Length

TL;DR

This paper explores the ambiguity in defining manifold topology at the Planck scale, demonstrating a smooth topology change via noncommutative geometries and discussing implications for D-brane theories in M(atrix) theory.

Contribution

It introduces a method to smoothly change topology at the Planck length using noncommutative geometries, with applications to D-branes in M(atrix) theory.

Findings

Topology can change smoothly at the Planck scale.

Explicit example of topology change from sphere to torus.

Implications for D-brane theory in quantum gravity.

Abstract

A basic arbitrariness in the determination of the topology of a manifold at the Planck length is discussed. An explicit example is given of a `smooth' change in topology from the 2-sphere to the 2-torus through a sequence of noncommuting geometries. Applications are considered to the theory of D-branes within the context of the proposed $M$(atrix) theory.