Nonabelian Poisson Manifolds from D-Branes

TL;DR

This paper develops a mathematical framework for nonabelian Poisson manifolds derived from D-branes, incorporating matrix-valued coordinates and their quantization, to model classical dynamics in string theory.

Contribution

It introduces a novel construction of C*-algebras associated with arbitrary simple Lie algebras and Poisson manifolds, extending classical mechanics to nonabelian, matrix-valued settings.

Findings

Defines a nonabelian Poisson manifold framework

Constructs C*-algebras for matrix-valued dynamics

Provides a basis for matrix-valued classical and quantum models

Abstract

Superimposed D-branes have matrix-valued functions as their transverse coordinates, since the latter take values in the Lie algebra of the gauge group inside the stack of coincident branes. This leads to considering a classical dynamics where the multiplication law for coordinates and/or momenta, being given by matrix multiplication, is nonabelian. Quantisation further introduces noncommutativity as a deformation in powers of Planck's constant. Given an arbitrary simple Lie algebra and an arbitrary Poisson manifold, both finite-dimensional, we define a corresponding C*-algebra that can be regarded as a nonabelian Poisson manifold. The latter provides a natural framework for a matrix-valued classical dynamics.