The main objective of the proposal is to develop methods of quantum field theory on algebraic curves, with applications to the study of complex-analytic and algebraic properties of moduli spaces --- the problems which are formulated using classical mathematics. The goals of the complex-analytic part of the project are the following: 1) derivation of the higher genus analog of Kronecker's second limit formula; 2) study of a new Kaehler metric on moduli spaces of punctured Riemann surfaces (so-called Takhtajan-Zograf metric), and Chern forms on moduli spaces of parabolic vector bundles; 3) application of the Weil-Petersson geometry of the universal Teichmueller space for the study of the univalent functions in the unit disk; 4) new relation between solution of the Riemann-Hilbert problem and the Wess-Zumino-Witten theory; 5) development of non-perturbative methods in geometric approach to quantum Liouville theory. This will result in major advances in the areas of mathematical physics related to the complex geometry of moduli spaces of Riemann surfaces and vector bundles, will give a remarkable product formula for Riemann theta-function in higher genus, will provide a new understanding of the classical Riemann-Hilbert problem, and will introduce new infinite-dimensional methods in the theory of the univalent functions.
The immediate and long term goals of the algebraic side are the following: 1) unification of classical theory of abelian differentials on Riemann surfaces with the adelic approach of C.~Chevalley and A.~Weil --- formulation of ``differential and integral calculus'' on algebraic curves over algebraically closed ground field, to be used for constructing quantum field theories on algebraic curves and for reciprocity laws; 2) development of ``algebraic harmonic analysis'' on adeles when the corresponding residue class field is not a finite field. This will result in a dramatic advance of our understanding of ``calculus'' on algebraic curves with applications that will possibly include fields of algebraic numbers.
The last thirty years have been characterized by remarkable interaction between mathematics and physics, which let to dramatic discoveries in various mathematics disciplines, bringing together the areas that were thought to be apart. In many cases application of methods of quantum fields and strings has been very successful, once again justifying Riemann's old idea of ``probing'' mathematical objects with physical theories and translating the physical ``output'' back into the realm of mathematics. When probing mathematical objects by quantum field theory, the mathematical output is encoded in partition function and correlation functions of quantum fields, defined by Feynman path integral, usually given in terms of perturbation expansion, and expressed through the critical value of the classical action and so-called ``quantum corrections''. Symmetries of the theory manifest themselves in Ward identities --- fundamental relations between partition function and correlation functions, which in many cases produce new and unexpected mathematical results. Proper understanding and exploitation of these symmetries is in the heart of today's applications of quantum physics to different mathematical disciplines. The realization of the main goals of the proposal will significantly contribute to the fundamental interface between mathematics and physics. This applies to the complex-analytic part as well as to the algebraic part; the fundamental properties of the latter are discrete, 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 this analogy further by formulating an approach based on the algebraic version of a Green's function, which is a fundamental object of classical analysis and mathematical physics, and is extensively used in quantum field theory.
