Quantum Dynamics on the Worldvolume from Classical su(n) Cohomology

TL;DR

This paper demonstrates that the Lie algebra of volume-preserving diffeomorphisms on certain compact Kähler manifolds can be approximated by su(n) as n approaches infinity, linking classical Nambu brackets to su(n) cohomology.

Contribution

It establishes a rigorous connection between classical worldvolume symmetries and their quantum algebraic approximations using su(n) in the context of p-branes.

Findings

su(n) approximates volume-preserving diffeomorphisms as n→∞

Classical Nambu brackets are quantized by su(n) cocycles

Provides a mathematical framework for quantum p-brane symmetries

Abstract

A key symmetry of classical $p$-branes is invariance under worldvolume diffeomorphisms. Under the assumption that the worldvolume, at fixed values of the time, is a compact, quantisable K\"ahler manifold, we prove that the Lie algebra of volume-preserving diffeomorphisms of the worldvolume can be approximated by $su(n)$, for $n\to\infty$. We also prove, under the same assumptions regarding the worldvolume at fixed time, that classical Nambu brackets on the worldvolume are quantised by the multibrackets corresponding to cocycles in the cohomology of the Lie algebra $su(n)$.