A noncommutative sigma-model

We begin to study a sigma-model in which both the space-time manifold and the two-dimensional string world-sheet are made noncommutative. The most precise results apply to the case where both the space-time manifold and the two-dimensional string world-sheet are replaced by noncommutative 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.