Crystalline Cohomology of Superschemes

We introduce a notion of crystalline cohomology for superschemes and show that it is isomorphic to the usual crystalline cohomology of the underlying commutative scheme if 2 is invertible on the base. Wess-Zumino terms play a crucial role in the matching of theoretical predictions in the framework of Quantum Chromodynamics of the decay of the neutral pion π into two photons π −→ γ + γ and the experimentally observed frequency. Wess-Zumino terms are often known to correspond to suitable cohomology groups related to de Rham cohomology of the target space of a suitable Σ-model. There are situations of physical interest where the target space is not a commutative manifold but rather a superspace. For example, consider the Green-Schwarz Wess-Zumino term in the superstring action as discussed in [10]. This concerns a Σ-model quantum field theory with maps φ : Σ −→M super where Σ is a surface and M super is a certain supermanifold whose associated commutative manifold is simply R for a suitable m. The Wess-Zumino term is given by the integral