Abstract:We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler-Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calculated explicitly. The gravitino terms pose additional analytic difficulties to show smoothness of its weak solutions which are overcome using Rivière's regularity theory and Riesz potential theory.

We discuss the $\zeta-$regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.Comment: LaTeX, 18 page

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of non-linear super symmetric sigma models, the action functional underlying string theory, can be obtained from a geometric action functional on super Riemann surfaces. All invariances of the super symmetric action functional are explained in super geometric terms and the action functional is a functional on the moduli space of super Riemann surfaces.

We prove a generalized version of the well-known Lichnerowicz formula for the square of the most general Dirac operator D on an even-dimensional spin manifold associated to a metric connection ∇. We use this formula to compute the subleading term Φ 1 (x, x, D 2 ) of the heat-kernel expansion of D 2 . The trace of this term plays a key-rôle in the definition of a (euclidian) gravity action in the context of non-commutative geometry. We show that this gravity action can be interpreted as defining a modified euclidian Einstein-Cartan theory. ⋆ Supported by the European Communities, contract no. ERB 401GT 930224 expansion, Wodzicki residue, gravity for the most general Dirac operator D associated to a metric connection ∇ on a compact spin manifold M with dim M = 2n ≥ 4. We proceed as follows: According to the main theorem of [KW], there is a relation between the Wodzicki residue Res(△ −n+1 ) of a generalized laplacian△ on a hermitian bundle E over M and