A noncommutative sigma-model

We begin to study a sigma-model in which both the spacetime manifold and the two-dimensional string world-sheet are made noncommutative. We focus on the case where both the spacetime manifold and the two-dimensional string world-sheet are replaced by noncommutative 2-tori. In this situation, we are able to determine when maps between such noncommutative tori exist, to derive the Euler–Lagrange equations, to classify many of the critical points of the Lagrangian, and to study the associated partition function.Varghese Mathai and Jonathan Rosenber