Noncommutative Geometry, Supergeometry, Gauge Theory and M-Theory

University Of California-Davis, Davis CA

Investigators

Abstract

Abstract - DMS 0204927 The main goal of the proposal is to apply methods of noncommutative geometry and supergeometry to the mathematical problems arising in theoretical physics. The development of (super)string theory led to the idea that this theory should by embedded into more general theory called M-theory. This hypothetical theory should live in 11-dimensional space and the corresponding low energy action should be given by 11-dimensional supergravity. At this moment a rigorous definition of M-theory is not known. Nevertheless, using various heuristic tools (first of all, dualities) physicists have created a beautiful and consistent picture, where all versions of (super)string theory can be obtained as limiting cases.A promising approach to a rigorous definition of M-theory is based on so called M(atrix) theory. The action functional of BFSS M(atrix) model can be obtained from ten-dimensional supersymmetric Yang-Mills theory ( 10 D SYM theory) by means of dimensional reduction to (0+1)-dimensional space.(This means that we should consider only gauge fields that do not depend on spatial variables, but can be time-dependent.) It was shown by A.Connes, M. Douglas and A. Schwarz (1998) that compactification of M(atrix) theory can lead to gauge theories on noncommutative spaces, in particular, on noncommutative tori.This paper opened the way for application of Connes' noncommutative geometry to string/M-theory . It became very popular, especially after the appearance of the influential Seiberg-Witten paper (1999) , that contains together with important new results, the understanding of Nekrasov-Schwarz noncommutative instantons (1998), gauge (complete) Morita equivalence defined by A.Schwarz (1998),and Pioline-Schwarz background independence (1999) from the string theory viewpoint. The number of papers considering the relation of string/M-theory to noncommutative spaces grows exponentially (today it is close to a thousand); many of these articles were influenced by the papers of A. Schwarz and his collaborators. A. Schwarz intends to continue his work on applications of noncommutative geometry to string/ M-theory. He is planning to analyze thoroughly the general theory of gauge fields on noncommutative spaces and its relation to the duality in physics. He would like to apply (in collaboration with M. Movshev) this general theory to the construction and analysis of gauge theories on noncommutative curved spaces. One more direction the PI would like to pursue ( also together with M. Movshev) is a search of formulation M(atrix) theory ( and, more generally, maximally supersymmetric gauge theory) in a manifestly supersymmetric form. The solution of this problem is interesting by itself, but M. Movshev and A. Schwarz are planning to use it also as a starting point in the construction of a maximally supersymmetric Dirac-Born-Infeld action for nonabelian gauge fields and on noncommutative spaces. A. Schwarz is planning to develop a complex version of Connes' noncommutative geometry and the theory of noncommutative theta functions having in mind possible applications to noncommutative instantons and other BPS fields.He aims also to study the role of K-theory in string/ M-theory. M. Movshev is planning to analyze algebraic structures arising in open string field theory. It is clear now that noncommutative geometry is quite useful in physics. The PI intends to develop methods of noncommutative geometry and to apply these methods to various problems arising in string/ M-theory. It seems that noncommutative geometry should play an important role in rigorous formulation of M-theory. The PI hopes that his work will contribute to the search of appropriate mathematical formalism.