caa a22 4500 475839668 CHVBK 20180406123858.0 cr unu---uuuuu 170329e20001101xx s 000 0 eng 10.1023/A:1010685903329 doi (NATIONALLICENCE)springer-10.1023/A:1010685903329 Prástaro Agostino Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ‘La Sapienza', Via A. Scarpa, 16-00161, Rome, Italy, e-mail aut (Co)bordism Groups in Quantum PDEs [Elektronische Daten] [Agostino Prástaro] 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 Prástaro 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, Êk ⊂ 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. Kluwer Academic Publishers, 2000 partial differential equations on noncommutative manifolds nationallicence (co)bordism in noncommutative manifolds nationallicence integral (co)bordisms in quantum PDEs nationallicence conservation laws and quantum Hopf algebras nationallicence quantum supergravity nationallicence quantum superstrings nationallicence Acta Applicandae Mathematica Kluwer Academic Publishers 64/2-3(2000-11-01), 111-217 0167-8019 64:2-3<111 2000 64 10440 https://doi.org/10.1023/A:1010685903329 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1010685903329 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Prástaro Agostino Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ‘La Sapienza', Via A. Scarpa, 16-00161, Rome, Italy, e-mail aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers 64/2-3(2000-11-01), 111-217 0167-8019 64:2-3<111 2000 64 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer