Principal Investigator: Albert Schwarz

The first of these projects concerns applications of p-adic methods to topological strings. The goal of these projects is to express physical quantities in terms of arithmetic geometry over p-adic numbers and to use these expressions to prove integrality theorems. Another set of investigations deals with supersymmetric deformations of quantum field theories. This is joint work with M. Movshev and applies the theory of L-infinity and A-infinity algebras. Another project is devoted to the analysis of the relation between string theory and quantum field theory. The principal investigator has argued that quantum field theory can be formulated such that time and space do not appear as primary notions, and is planning to show that string theory can be regarded as a quantum field theory in this sense.

These research projects are based on the application of methods of modern mathematics to physics, particularly making use of arithmetic geometry and homological algebra to improve our understanding of quantum field theory and string theory. Sophisticated mathematics originally developed for other purposes such as the study of Diophantine equations in number theory seems to bear directly on challenging issues in theoretical physics.

Project Report

My project is an interdisciplinary project on the interface between mathematics and physics. In recent decades the renewed collaboration between mathematics and physics led to spectacular results in both fields. Quantum field theory and string theory essentially use the results of many branches of modern mathematics. From the other side the ideas borrowed from physics transformed several mathematical fields, like low-dimensional topology and algebraic geometry. I have participated actively in this development, both as a mathematician and as a theoretical physicist. (Topologically non-trivial solutions of equations of motion - monopoles and instantons, first examples of topological quantum field theories, applications of noncommutative geometry, etc .) My recent results go in the same direction. They are based on the application of arithmetic geometry and homological algebra to physics. In one of my papers that served as a basis of my proposal I developed some basic notions of "physics over a ring" where the role of real or complex numbers play elements of some ring. The physics over a ring can have fundamental meaning if physical quantities take only discrete values (for example if length can take only values proportional to some unit length). In any case it gives useful technical tools allowing to obtain new results in conventional physics. In particular, in my papers with V. Vologodsky and J. Walcher " physics over p-adic numbers" was used to obtain some integrality theorem in the theory of topological strings. Homological algebra enters physics in the framework of BRST formalism. Among the results of my project was calculation of some homology groups that appear in many problems of quantum field theory and string theory (together with M. Movshev, Renjun Xu and A. Mikhailov). In a series of papers with Jia-Ming Liou we have solved some topological problems that can be useful in the construction of nonperturbative string theory. (It was conjectured that such a theory can be constructed in terms of infinite-dimensional Grassmannian; we have studied the topological properties of the Krichever map relating moduli spaces of algebraic curves with Grassmannian.) My recent papers are devoted to quantum curves, that appear in many questions of theoretical physics and mathematics (matrix models, topological strings, knot theory etc). I define quantum curves and discrete quantum curves as pairs of differential or difference operators satisfying some equations; I describe moduli spaces of these curves.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Christopher W. Stark
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University of California Davis
United States
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