Why Matrix theory works for oddly shaped membranes

TL;DR

This paper provides a straightforward proof of why matrix theory can approximate membranes of arbitrary Riemann surfaces, highlighting limitations for noncompact membranes and implications for string theories.

Contribution

It offers a simple proof of matrix approximation for membranes of any Riemann surface and discusses its limitations in certain string theories.

Findings

Noncompact membranes cannot be approximated by matrices.

Poisson algebra on any compact phase space is U(infinity).

Matrix approximation fails in theories lacking conserved 3-form charge.

Abstract

We give a simple proof of why there is a Matrix theory approximation for a membrane shaped like an arbitrary Riemann surface. As corollaries, we show that noncompact membranes cannot be approximated by matrices and that the Poisson algebra on any compact phase space is U(infinity). The matrix approximation does not appear to work properly in theories such as IIB string theory or bosonic membrane theory where there is no conserved 3-form charge to which the membranes couple.