M(atrix) Theory on an Orbifold and Twisted Membrane

TL;DR

This work presents a nonperturbative formulation of M-theory on the orbifold $S_1/{\mathbf Z}_2$ and analyzes the resulting heterotic M(atrix) theory. Through Chan-Paton constraints and area-preserving diffeomorphism, it identifies allowable gauge groups $${\mathbf SO}}(2N), {\mathbf SO}}(2N+1), {\mathbf USp}(2N)$$ and shows that open two-branes are permitted only for $${\mathbf SO}}(2N)$$ and $${\mathbf SO}}(2N+1)$$, with twisted sectors required to cancel local cosmological constants and worldsheet anomalies. The twisted sector supplies 16 fundamental spinors per fixed point, enabling heterotic string-like spectra and a consistent large-$N$ limit that yields open and closed two-brane configurations consistent with dual Type IA and heterotic string theories. In the large-$N$ limit, matrix configurations map to continuum APD algebras, revealing explicit correspondences between brane topologies (disk, cylinder, Möbius, Klein bottle, RP2) and gauge groups. These results provide a framework for understanding gauge symmetry enhancement and twisted-sector dynamics in strongly coupled heterotic M-theory and its dual string theories.

Abstract

M(atrix) theory on an orbifold and classical two-branes therein are studied with particular emphasis to heterotic M(atrix) theory on $S_1/Z_2$ relevant to strongly coupled heterotic and dual Type IA string theories. By analyzing orbifold condition on Chan-Paton factors, we show that three choice of gauge group are possible for heterotic M(atrix) theory: SO(2N), SO(2N+1) or USp(2N). By examining area-preserving diffeomorphism that underlies the M(atrix) theory, we find that each choices of gauge group restricts possible topologies of two-branes. The result suggests that only the choice of SO(2N) or SO(2N+1) groups allows open two-branes, hence, relevant to heterotic M(atrix) theory. We show that requirement of both local vacuum energy cancellation and of worldsheet anomaly cancellation of resulting heterotic string identifies supersymmetric twisted sector spectra with sixteen fundamental representation spinors from each of the two fixed points. Twisted open and closed two-brane configurations are obtained in the large N limit.