Geometric Aspects of Quantum Field Theory and Topological String

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Under this award, the principal investigator will continue his research on the quantum geometry of complex varieties. Further, in collaboration with the co-PI, the investigator will undertake a study of certain algebraic structures arising from quantum field theory and topological strings. The research contains two major perspectives. The first aim is to develop the mathematical foundations of the quantum geometry of the B-twisted topological string on a Calabi-Yau geometry; a complete description will include open-closed string field theory and the Landau-Ginzburg B-model. Secondly, the PI and co-PI will analyze the perturbative quantization of two dimensional sigma models and develop various algebraic index theories for higher differential operators via a rigorous observable theory. The blossoming of quantum field theory and string theory has provided a great driving force in the creation of new areas in mathematics. The proposed project explores examples arising from quantum field theory relevant to geometry and topology in a mathematically rigorous setting. The aim is to develop new mathematical tools and reveal the geometry of quantum phenomenon by anchoring physical intuition and arguments in the harbor of mathematics. The implications of the proposed projects live both in an array of mathematical fields (differential geometry, algebraic geometry, topology) and theoretical physics.