Co)bordism Groups in Quantum PDEs

Abstract

In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric theory of quantum PDEs (QPDEs). In particular, a criterion of formal integrability is given that extends to QPDEs previously obtained by D. C. Spencer and H. Goldschmidt for PDEs for commutative manifolds, and by Prastaro for super PDEs. Quantum manifolds are seen as locally convex manifolds where the model has the structure A m 1 1×···×A m s s , with A≡A 1×···×A s a noncommutative algebra that satisfies some particular axioms (quantum algebra). A general theory of integral (co)bordism for QPDEs is developed that extends our previous for PDEs. Then, noncommutative Hopf algebras (full quantum p-Hopf algebras, 0≤p≤m−1) are canonically associated to any QPDE, Ek ⊂ Ĵk m(W) whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, we carefully study QPDEs for quantum field theory and quantum supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on spacetime, where A is a quantum algebra. For such equations, canonical quantizations are obtained and the quantum and integral bordism groups and the full quantum p-Hopf algebras, 0≤p≤3, are explicitly calculated. Then, the existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved.