Quantum geometry of super PDEs
type: Item
instance of: scholarly article
title: Quantum geometry of super PDEs
zbMATH Open document ID: 0887.58064
DOI: 10.1016/0034-4877(96)88921-X
published in: Reports on Mathematical Physics
publication date: 1998-05-10T00:00:00Z
review text: The body of this paper contains Introduction, Appendix and the following three sections: 1. Super partial differential equations; 2. Quantization of super PDEs; 3. Geometry of quantum supergravity. In the first section the author develops a geometric theory for super PDEs by introducing supermanifolds modelled on locally convex spaces, where the model is \(A^{m,n}\equiv A^m_0\times A^n_1\), and \(A\) is a Banach-Grassmann superalgebra. This approach allows him to consider in an algebraic way superdual spaces and superderivative spaces and to develop a formal theory for super PDEs. Then he obtains existence theorems of formal integrability, formal prolongations, obstructions to solvability of Cauchy problems, characterization of singular solutions, conservation laws, and so on. In the second section the author develops the algebraic meaning of super PDEs in a categorial context. He relates the concept of singular solutions for super PDEs to superdistributional solutions of affine super PDEs. Then a generalized Morse theory is used to give a geometric description of the quantum tunneling effects in super PDEs. Previous results of Witten and Atiyah in topological field theories are explicitly extended to the geometric framework for super PDEs. In the third section the author develops the theory of non-holonomic jet-superderivative spaces and non-holonomic superconnections. These structures are necessary to give a geometric formulation of supergravity. Then the previous machinery is applied to a geometrical formulation of quantum supergravity. In the Appendix the author recalls some fundamental definitions and results on the spectral theory of algebras.
reviewed by: Viktor Abramov
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 35Q05
Mathematics Subject Classification ID: 58C50
Mathematics Subject Classification ID: 83E50
zbMATH DE Number: 1090407
zbMATH Keywords: super partial differential equations (super PDE)
zbMATH Keywords: supermanifolds
zbMATH Keywords: supergravity
zbMATH Keywords: topological field theory
zbMATH Keywords: super bundles
zbMATH Keywords: super connections
zbMATH Keywords: Cartan superdistribution
author: Agostino Prástaro
MaRDI profile type: Publication
cites work: Fields, observables and gauge transformations. II
cites work: Q3910965
cites work: Q4003853
full work available at URL: 0034-4877(96)88921-x
OpenAlex ID: W2078707866
description: scientific article; zbMATH DE number 1090407