Supersymplectic geometry of supersymmetric quantum field theories

Abstract

We develop a conceptually new, geometric approach to supersymmetry. In particular, we argue that the construction of a generic supersymmetric theory entails only symplectic geometry either in a loop space parametrized by the bosonic degrees of freedom or in a superloop space parametrized by both bosonic and fermionic degrees of freedom. In the bosonic loop space a generic supersymmetry theory can be constructed using a model dependent loop space symplectic two-form, the corresponding symplectic one-form and a model independent vector field that determines circle action in the loop space. In the superloop space the construction of a generic supersymmetric theory employs a model independent symplectic two-form, the pertinent symplectic one-form, a model independent vector field that determines circle action in the superloop space, and the interaction is obtained by introducing a canonical transformation in the superloop space. A Poincaré supersymmetric quantum field theory is a realization of our formalism in terms of space-time variables that admit a natural Lorentz-invariant interpretation. We expect that our geometric approach to supersymmetry opens a novel point of view to a large class of problems, including the mechanism of supersymmetry breaking, structure of topological field theories and even aspects of quantum integrability.