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

Abstract

This article provides a brief discussion of the functional of super Riemann surfaces from the point of view of classical (i.e., not "super-") differential geometry. The discussion is based on symmetry considerations and aims to clarify the "borderline" between classical and super differential geometry with respect to the distinguished functional that generalizes the action of harmonic maps and is expected to play a basic role in the discussion of "super Teichmüller space". The discussion is also motivated by the fact that a geometrical understanding of the functional of super Riemann surfaces from the point of view of super geometry seems to provide serious issues to treat the functional analytically.

Symmetries and conservation laws of a nonlinear sigma model with gravitino

“…The major reason for the introduction of those models was super symmetry. As was argued in [20] super symmetry requires anti-commutative variables. Hence a full super symmetry cannot be expected for the action functional (5).…”

“…The main motivation for the introduction of the two-dimensional nonlinear supersymmetric sigma model in quantum field theory, or more specifically super gravity and super string theory, are its symmetries, see for instance [3,9]. Furthermore, as argued in [20], the functional is determined by its symmetries together with suitable bounds on the order of its Euler-Lagrange equations. While super symmetric models are usually formulated using anti-commutative variables, in [18] an analogue of the two-dimensional nonlinear supersymmetric sigma model using only commutative variables was introduced.…”

Energy quantization for a nonlinear sigma model with critical gravitinos

“…To show (25), it may be more intuitive to transform them to a cylinder. Let (r k , θ k ) be the polar coordinate around x k .…”

“…However, because of the various spinor fields involved, new difficulties arise. The geometric aspects have been developed in mathematical terms in [25], but this naturally involves anti-commuting variables which are not amenable to inequalities, and therefore variational methods cannot be applied, and one rather needs algebraic tools. This would lead to what one may call super harmonic maps.…”

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

“…Nevertheless, as described, our functional still possesses many symmetries, and these are crucial for its analysis and for its geometric content. Furthermore, as described in [58], the functional A is completely determined by the requirement of rescaled conformal and super Weyl invariance, given that the equations of motion are at most of second order.…”