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Moduli identities and cycles cohomologies by integral transforms in derived geometry. (English) Zbl 06678567
Summary: Generalizations of Derived categories and their deformed versions are used to develop a theory of ramifications of field studied in the geometrical Langlands program to obtain the correspondences between moduli stacks and solution classes in field theory, represented cohomologically under several versions of generalized Penrose transforms on cycles whose Spec has objects in a quantum algebra whose derived category is an extension of holomorphic bundles categories with a special connection (Deligne connection). The co-cycles in this spectrum can conform the Langlands correspondence via the Penrose transforms on generalized D-modules in moduli stacks defined on adequate holomorphic vector bundles and their possible extension to meromorphic connections, as an example. In this correspondence problem a Zuckerman functor is a factor of the universal functor of derived sheaves of Harish-Chandra which can be worked widely in the Langlands geometrical program to the mirror symmetry in different physical stacks of the Universe (for example, worked in different moduli spaces identities to different moduli problems).
One important cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend them to gauge solutions to other fields using the topological groups symmetries that define their interactions (generalized Verma modules). For it, are analyzed the cohomological groups that can establish a theory of the Ext functor to characterizing of a twisted derived category and their elements as ramifications of a field (to the field equations) and followed through the application of the corresponding Yoneda algebra where is searched extends the action to an endomorphism of Verma modules of critical level bundles through the action of a Lie algebra and on a cohomological space of zero dimension, which we want, that is to say; the first member of the Penrose transform must be isomorphic to a cohomology group of zero dimension on the Verma modules belonging to a twisted derived category whose points are Hecke sheaves, but that in our spectral resolution must be at least of $$q = 1$$, dimension.
##### MSC:
 14D24 Geometric Langlands program (algebro-geometric aspects) 83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism 53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category 11R39 Langlands-Weil conjectures, nonabelian class field theory