Proposal: DMS-9802574 Principal Investigator: Leon Takhtajan
The project is devoted to the study of quantum field theories on algebraic curves from complex analytic and algebraic points of view. In the complex analytic setting the main goal is to develop a geometric approach for calculating correlation functions of degenerate primary fields in the two-dimensional quantum gravity and to apply the chiral action functional for higher genus Riemann surfaces to the differential geometry of Earle-Eells principal fiber bundle over the Teichmuller space and to other problems. In the algebraic setup the goal is to develop, following an idea of E. Witten, a new method in the theory of algebraic functions of one variable over arbitrary field of constants using various quantum field theories over algebraic curves. In particular, for the theory of free bosons this "field-theoretical" paradigm will provide a new proof of A. Weil's reciprocity law on algebraic curves. Development of this program will have a broad impact on the class field theory of fields of algebraic functions with a possibility to work for algebraic number fields as well. In the latter case the goal is to derive classical Gauss quadratic reciprocity law and its generalizations as "conformal Ward identities" for quantum field theories.
Ideas from physics, originated from the study of surrounding world, have always played a pivotal role in the development of mathematics. Recently dramatic progress was made in the study of geometry and topology in four dimensions by using methods of quantum field theory. In a sense, special quantum fields on a four-dimensional manifold - a geometric object in four-dimensional space - turned out to probe its geometric and topological properties and by measuring their response one gets a nontrivial information about the manifold. The goal of this proposal is to develop a similar method for studying algebraic and arithmetic properties of surfaces and number fields (like the usual field of rational numbers) by using quantum fields. Many fundamental properties of these objects are discrete and so it is natural to study them using quantum theory. In particular, many classical 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 field theory. The main goal of the proposal is to develop a new paradigm based on quantum theory for studying the 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.
