Quantum Field Theories on Algebraic Curves and Applications

DMS-0204628 - Leon Takhtajan Abstract The proposal is devoted to the study of quantum field theories on algebraic curves from complex analytic and algebraic points of view. String theory is the unifying theme of the complex-analytic part of the project. The main goals of this part are: 1) geometric formulation of two-dimensional quantum Toda theories associated with simple Lie algebras; 2) Fermi-Bose correspondence for compact Riemann surfaces and closed smooth Jordan curves; 3) proof of general Kleinian reciprocity on deformation spaces; 4) construction of the universal Weil-Petersson potential for the universal Teichmuller space; 5) curvature properties of the new Kahler metric on moduli spaces of punctured Riemann surfaces; 6) explicit form of factorization formula for determinants of Laplace operators acting on higher order differentials; 7) Faddeev-Popov ghosts for string field theory of open strings. Symmetries and Ward identities for correlation functions are guiding principles of the algebraic part of the project. The immediate and long term goals of this part are: 1) proof of adelic Fermi-Bose correspondence for vertex operators, complete construction of quantum field theory of multiplicative bosons and proof of A. Weil's reciprocity law; 2) construction of quantum field theories on algebraic curves in non-zero characteristic with Artin-Schreier and Kummer theories as main examples, proof of Artin's reciprocity law using quantum field-theoretical methods; 3) adelic formulation of the WZW theory on algebraic curves in characteristic zero. In mathematical development a pivotal role is played by ideas from physics, originated in the study of the surrounding world. The last twenty five years have been characterized by dramatic success in application of quantum fields and strings to different areas of mathematics. The idea is to probe mathematical objects by quantum theories and to measure their response in order to get new information about the various mathematical properties of these objects. Usually the output is encoded in terms of partition function and correlation functions of physical theory, and the problem is to decode it back in mathematical terms. The goals of this proposal are the following: 1) to extend further quantum-theoretic method for the study of complex analytic properties of surfaces in two dimensions and their families; 2) to develop a new systematic quantum-theoretic approach for arithmetic surfaces and algebraic number fields (like the usual field of rational numbers). Fundamental properties of these objects are discrete and it is only natural to study them at a microscopic scale using quantum theory. In particular, many classical mathematical results about fields of algebraic functions and algebraic numbers, known as reciprocity laws, can be interpreted as conservation laws (like conservation of energy) in quantum theory. The main goal of the algebraic part of the project is to develop a new approach based on quantum theory towards fundamental laws that are satisfied by the very basic mathematical objects: algebraic numbers - solutions of algebraic equations with coefficients being rational numbers, and algebraic functions - solutions of algebraic equations with coefficients being rational functions. Realization of the goals of the proposal will contribute to the fundamental interface between mathematics and physics.