Quaternions and M(atrix) theory in spaces with boundaries

TL;DR

This paper proposes a matrix model formulation of M-theory on spaces with boundaries, utilizing quaternionic and real matrix elements, and demonstrates membranes correctly end on the boundary within this framework.

Contribution

It introduces a novel matrix model approach for M-theory with boundaries, employing quaternionic and real matrices, and details how membranes interact with the boundary.

Findings

Matrix elements become real or quaternionic due to symmetry modding.

The gauge symmetry reduces from U(2N) to O(2N) or USp(2N).

Membranes are shown to end correctly on the boundary.

Abstract

A proposal for the matrix model formulation of the M-theory on a space with a boundary is given. A general machinery for modding out a symmetry in M(atrix) theory is used for a Z_2 symmetry changing the sign of the X_1 coordinate. The construction causes the elements of matrices to be equivalent to real numbers or quaternions and the symmetry U(2N) of the original model is reduced to O(2N) or USp(2N)=U(N,H). We also show that membranes end on the boundary of the spacetime correctly in this construction.