Open and Winding Membranes, Affine Matrix Theory and Matrix String Theory

TL;DR

This paper explores the structure of open and winding membranes in M-theory, revealing how their boundary conditions lead to affine algebra representations and providing a microscopic derivation of matrix string theory and related models.

Contribution

It introduces a novel approach to membrane truncation using affine algebra representations, connecting membrane boundary conditions to matrix string theory derivation.

Findings

Linear terms in mode expansion act as derivations on volume-preserving diffeomorphisms.

Consistent truncation occurs to affine algebra representations when one membrane direction is stretched.

Derivation of matrix superstring theory for parallel M5-branes from supermembrane truncation.

Abstract

We examine the structure of winding toroidal and open cylindrical membranes, especially in cases where they are stretched between boundaries. Non-zero winding or stretching means that there are linear terms in the mode expansion of the coordinates obeying Dirichlet boundary conditions. A linear term acts as an outer derivation on the subalgebra of volume-preserving diffeomorphisms generated by single-valued functions, and obstructs the truncation to matrix theory obtained via non-commutativity with rational parameter. As long as only one of the two membrane directions is stretched, the possible consistent truncation is to coordinates taking values in representations of an affine algebra. We show that this consistent truncation of the supermembrane gives a precise microscopic derivation of matrix string theory with the representation content appropriate for the physical situation. The matrix superstring theory describing parallel M5-branes is derived. We comment on the possible applications of the construction to membrane quantisation in certain M-theory backgrounds.