Deformed gauge theories

Abstract

In this lecture we discuss two possible ways of introducing gauge theories on non-commutative spaces. In the first approach the algebra of gauge transformations is unchanged, but the Leibniz rule changes (compared with gauge theories on commutative space). Consistency of equations of motion forces gauge field to be enveloping algebra-valued, which leads to new degrees of freedom. In the second approach we have to go to the enveloping algebra again if we want noncommu-tative gauge transformations to close in the algebra. However, no new degrees of freedom appear here because of the Seiberg-Witten map. This map enables one to express noncommutative gauge parameter and fields in terms of the corresponding commutative variables.