Quantum geometry of super PDEs

Abstract

In order to extend to super PDEs the theory of quantization of PDEs as contained in [53], we first develop a geometric theory for super PDEs (see also [54]). Superspaces and supermanifolds are introduced by using the concept of weak differentiability as usually given for locally convex spaces [23, 26]. This allows us to consider in algebraic way superdual spaces and superderivative spaces and to develop a formal theory for super PDEs that directly extends the previous ones for standard manifolds of finite dimension. In particular, we give a criterion of formal superintegrability for super PDEs, and show that a geometric theory of singular supersolutions, with singularities of Thom-Bordman type, can be formulated in the framework of super PDEs too. These results generalize the previous ones obtained for ordinary manifolds by H. Goldschmidt [17] and by Moscow's mathematical school [28, 29, 66–68, 75, 76]. Conservation superlaws associated to super PDEs are considered and related with some spectral sequences and wholly cohomological character of these equations. Then, the quantization of super PDEs is formulated on the ground of quantum cobordism [46, 53]. This is made in order to give an intrinsic and fully covariant geometric formulation of unified quantum field theory. In particular, a theory of quantum supergravity is developed. We explain how canonical quantization and quantum tunnelling effects arise in super PDEs. Furthermore, we explicitly extend previous results of Witten and Atiyah in topological quantum field theory [1, 2, 69–71] to our geometric framework for super PDEs. Obstructions to existence of quantum cobords in super PDEs are given by means of supercharacteristic classes. These results can be considered as a generalization of the recent results obtained by Gibbons and Hawking [16].