Objective
Givental has conjectured, that the U(1)-equivariant Floer cohomology of the universal covering of the loop space, of asymplectic manifold, should have the structure of a D-module, over the Heisenberg algebra of first order differential operators on a complex torus and that this should be the same as the quantum cohomology D-module of the manifold. I intent to study this conjectured equality and its implications in computing the quantum D-module. This implies also computation of the quantum ring, as the later is the semi-classical limit of the former. There are three concrete directions of research. First, note that there is a "Fourier transform" of equivariant cycles that transforms relations in the D-module to differential operators. If we could compute this transform, then we could compute the D-module. Because of the infinite dimensionality of the loop space though, there are problems with computing the integral involved I have managed to compute it in the case of positive toric manifolds, but this method relies on the Fourier expansion and doesn't seem to generalize to non-toric manifolds. For the case of general semi-positive symplectic manifolds, I propose A totally different method, which relies on using localization techniques and a certain exact sequence arising from Morse theory of the simplistic action functional, in order to regularize the ratios of relevant equivariant Euler classes. The second program I propose, is to use the model of Getzler, Jones and Petrack, for the equivariant cohomology of the loop space. They identify it with a version of the cyclic bar complex, involving Connes's operator B and this could be used to compute the relevant "Fourier transform". Thirdly, I intent to investigate applications of these, to Kontsevich's homological mirror symmetry proposal. I expect very valuable and necessary training benefit from my stay at the IHES. Professor Gromov is one of the pioneers and world experts in the field of geometric analysis (Gromov-Witten invariants) and this will be needed greatly in completing the construction of the module space of flow lines, involved in the first part of my project. Professors Kontsevichand Connes, could provide invaluable help in studying the cyclic bar complex, equipped with the Connes operator B, that I will use to compute the U(1) -equivariant cohomology of the universal cover of the loop space. Finally, since I intent to investigate applications of the previous work to professor Kontsevich's homological mirror symmetry conjecture, possible collaboration with him would be a great advantage. Besides these, in the IHES there is a very high level of interest in the relevant area of geometry and string theory. On my part, I look forward to contributing to this creative atmosphere with my expertise, enthusiasm and some new ideas.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics topology symplectic topology
- natural sciences mathematics pure mathematics algebra
- natural sciences physical sciences theoretical physics string theory
- natural sciences mathematics pure mathematics geometry
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Coordinator
91440 BURES-SUR-YVETTE
France
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