Representation Theory and Quantum Field Theory

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This proposal is centered around several examples of sheaves of vertex algebras that Malikov, Schechtman, Vaintrob, and Gorbounov discovered and investigated in the previous work. More specifically, the first part of the proposal has to do with computation of the algebras of global sections of these sheaves on Calabi-Yau manifolds. The known examples arising in the case of tori and projective spaces are promising. The second part is built on the recent discovery that a family of such vertex algebras arising in the case of the projective line is closely related to the super-algebra Lie sl(2|1) and to sl(2|1) - Toda field theory. The third is an attempt to generalize the relation of such sheaves to ellptic genera, quantum cohomology, and modular forms discovered in a joint work with Schechtman. This relation is visible through the fact that the character of the space of global sections of one such sheaf, a chiral de Rham complex, over over an even-dimensional projective space equals the equivariant signature of the corresponding loop space. The fourth is a project aiming at an explanation of a certain remarkable duality arising in the case of a simple Lie group, in physics terms if possible. That the whole proposal may be related to string theory is revealed, for example, in the fact that the above-mentioned chiral de Rham complex allows to reproduce the quantum cohomology of smooth projective toric varieties. The fifth is a separate project based on the previous work of I.Frenkel and Malikov; it has to do with a possible application of affine translation functors to BGG-type resolutions of admissible representations.This is a project simultaneously in the area of mathematics known as "Representation Theory," and in the area of mathematical physics called "Quantum Field Theory." Representation theory is generally considered to be the most useful area of mathematics for describing symmetries of all kinds, especially those that arise in nature. Quantum field theory may be thought of as the logical foundation for the quantum mechanics that govern the behavior of subatomic particles. An important source of inspiration for this project is string theory, which is widely accepted as the only candidate for a unified physical theory explaining all fundamental laws of Nature. The striking idea of string theory is that an elementary particle in physics can be described mathematically as a loop. The implications of this idea have made entire areas of mathematics previously thought of as totally abstract into everyday tools of research in theoretical physics. The specific goal of this project is to obtain a better understanding of the so-called "loop spaces" that arise in this point of view. Mathematical ideas from string theory have already found important applications in tomography, and surely there are more applications waiting to be found.