Membranes and Matrix Models

TL;DR

This paper explores membrane dynamics through matrix models, deriving a key Hamiltonian, analyzing zero-energy states, and examining solutions to matrix differential equations, contributing to the understanding of supersymmetric membrane theories.

Contribution

It provides a detailed derivation of a matrix Hamiltonian for membrane dynamics and investigates zero-energy states and solution spaces in supersymmetric matrix models.

Findings

Derived the Hamiltonian $H=- riangle - Tr \sum_{i<j}[X_i,X_j]^2$ for membrane models

Analyzed conditions for zero-energy bound states in SU(N)-invariant models

Explored solutions of matrix differential equations interpolating between $su(2)$ representations

Abstract

Section I contains introductory remarks about surface motions. Section II gives a detailed derivation of $H=-\Delta-Tr\sum_{i<j}[X_i,X_j]^2$ as describing a quantized discrete analogue of relativistically invariant membrane dynamics. Section III concerns the question of zero-energy bound-states in SU(N)-invariant supersymmetric matrix models. Section IV discusses the space of solutions of some differential matrix equations on $(-\infty,+\infty)$, interpolating between different representations of $su(2)$. Some exercises are added, and one remark/conjecture concerning 5-commutators.