Noncommutative Geometry, Supergeometry, Gauge Theory and M-Theory

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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 theorycalled M-theory. This hypothetical theory should live in11-dimensional space and the corresponding low energy actionshould be given by 11-dimensional supergravity. At thismoment a rigorous definition of M-theory is not known. Nevertheless, using various heuristic tools (first of all,dualities) physicists have created a beautiful andconsistent picture, where all versions of (super)stringtheory can be obtained as limiting cases.A promising approach to a rigorous definition of M-theory is based on socalled M(atrix) theory. The action functional of BFSS M(atrix) model can be obtained from ten-dimensional supersymmetricYang-Mills theory ( 10 D SYM theory) by means of dimensionalreduction to (0+1)-dimensional space.(This means that we should consider only gauge fields that do not depend on spatialvariables, but can be time-dependent.) It was shown by A.Connes, M. Douglas and A. Schwarz (1998) thatcompactification of M(atrix) theory can leadto 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 . Itbecame very popular, especially after the appearance of theinfluential Seiberg-Witten paper (1999) , that containstogether with important new results, the understanding ofNekrasov-Schwarz noncommutative instantons (1998), gauge(complete) Morita equivalence defined by A.Schwarz (1998),and Pioline-Schwarz background independence (1999) from thestring 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 thesearticles were influenced by the papers of A. Schwarz and hiscollaborators. A. Schwarz intends to continue his work onapplications 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 constructionand analysis of gauge theories on noncommutative curved spaces. One more direction the PI would like to pursue ( alsotogether with M. Movshev) is a search of formulationM(atrix) theory ( and, more generally, maximally supersymmetricgauge 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-Infeldaction for nonabelian gauge fields and on noncommutative spaces. A. Schwarz is planning to develop a complex version of Connes' noncommutative geometry and the theory ofnoncommutative theta functions having in mind possibleapplications to noncommutative instantons and other BPSfields.He aims also to study the role of K-theory in string/ M-theory. M. Movshev is planning to analyzealgebraic structures arising in open string field theory. It is clear now that noncommutative geometry is quite usefulin physics. The PI intends to develop methods ofnoncommutative geometry and to apply these methods tovarious problems arising in string/ M-theory. It seemsthat noncommutative geometry should play an important rolein rigorous formulation of M-theory. The PI hopes thathis work will contribute to the search of appropriatemathematical formalism.