Dirichlet branes and a cohomological definition of time flow

TL;DR

This paper develops a cohomological framework for describing the dynamics of Dirichlet branes with matrix-valued coordinates, introducing generalized Poisson brackets on universal enveloping algebras to handle nonabelian and noncommutative structures.

Contribution

It introduces a cohomological approach to define time flow for matrix-valued coordinates, extending classical Poisson structures to universal enveloping algebras for Dirichlet branes.

Findings

Generalized Poisson brackets on universal enveloping algebras are natural for matrix-valued coordinates.

The framework accommodates nonabelian and noncommutative geometries in brane dynamics.

Provides a mathematical foundation for quantizing matrix-valued coordinate systems.

Abstract

Dirichlet branes are objects whose transverse coordinates in space are matrix-valued functions. This leads to considering a matrix algebra or, more generally, a Lie algebra, as the classical phase space of a certain dynamics where the multiplication of coordinates, being given by matrix multiplication, is nonabelian. Further quantising this dynamics by means of a star-product introduces noncommutativity (besides nonabelianity) as a quantum h-deformation. The algebra of functions on a standard Poisson manifold is replaced with the universal enveloping algebra of the given Lie algebra. We define generalised Poisson brackets on this universal enveloping algebra, examine their properties, and conclude that they provide a natural framework for dynamical setups (such as coincident Dirichlet branes) where coordinates are matrix-valued, rather than number-valued, functions.