Quantum geometry of super PDEs
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.
super partial differential equations (super PDE)
scientific article; zbMATH DE number 1090407