Super Riemann surfaces, metrics and gravitinos

Abstract

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.

The Functional of Super Riemann Surfaces – A “Semi-Classical” Survey

“…Although much is known on this matter, already, it still seems worth presenting a detailed account on how supersymmetry enforces super geometry. In fact, our discussion should be understood as being complementary to what has been presented in [8] and, in particular, in [9].…”

“…This is rather close, indeed, to super geometry. Indeed, one can show that the action functional A SRS from (39) is equivalent to the action functional I SDH -studied in [8] under the assumption of a trivial family of super Riemann surfaces. It seems unavoidable to use the full language of super geometry, in particular ringed spaces involving anti commutative variables, and reminiscent to the supersymmetry transformations, which seem to have a clear geometrical interpretation only within the realm of super geometry.…”

“…Yet, one may pose the question to what extend the functional (6) and super Riemann surfaces can also be motivated within the setting of "classical" (i.e., non-super) differential geometry. In fact, in [8], it is discussed how super Riemann surfaces are related to spinor bundles over Riemann surfaces together with the gravitino field χ . This classical geometrical background seems more suitable for geometrical analysis than the super setting.…”

“…The formal similarity between (13) and (1) is apparent. Within the super setting, the supersymmetry transformations (8)- (11) have the geometrical meaning of a specific (infinitesimal) diffeomorphism on the super Riemann surface M. For details, we refer to [8] and, especially, to [9]. There, it is shown, for the first time, in all details that the functional (13) is indeed well-defined on the moduli space of super Riemann surfaces like (1) is well-defined on Teichmüller space.…”

From Harmonic Maps to the Nonlinear Supersymmetric Sigma Model of Quantum Field Theory: at the Interface of Theoretical Physics, Riemannian Geometry, and Nonlinear Analysis

“…2 We use the sign conventions of [34], i.e., For the moment, we assume χ to be smooth, but smoothness will become an issue below. In [35], a gravitino is introduced as a section of the bundle S ⊗ T * M ; we can, however, use the metric γ to identify T * M with T M .…”