Noncommutative Gauge Theories in Matrix Theory

TL;DR

Matrix theory compactifications on flat quotients $R^n/\Gamma$ generically produce noncommutative gauge theories described by twisted group algebras $C^{\alpha}\Gamma$ associated with a projective regular representation. The authors derive the general quotient-solution structure, showing that physical gauge data live on the dual space and that the surviving gauge algebra is the commutant of the twisted group algebra; global symmetries arise from $H^1(\Gamma,U(1))$. The paper provides concrete realizations via (i) quantum tori, where a noncommutative star-product yields deformed Yang–Mills, and (ii) ALE orbifolds, where the gauge sector aligns with extended Dynkin diagrams and bifundamental hypermultiplets, with orientifolds treated through a $Z_2$ grading. This framework unifies Matrix theory compactifications with noncommutative geometry and clarifies how orientifolds and other discrete symmetries shape the resulting NC gauge theories, with potential implications for string/M-theory compactifications and dualities.

Abstract

We present a general framework for Matrix theory compactified on a quotient space R^n/G, with G a discrete group of Euclidean motions in R^n. The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We characterize the resulting noncommutative gauge theory in terms of the twisted group algebra of G associated with a projective regular representation. Also we show how to extend our treatments to incorporate orientifolds.