Algebraic Geometry of Difference Operators and Real Bundles

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DMS-0401448 David Ben-ZviThe geometric Langlands program proposes an extraordinary organizing principle for representation theory over algebraic curves, inspired by the Langlands philosophy which binds harmonic analysis and Galois theory over number fields. Namely, it describes a geometric analog of spectral theory on moduli spaces of bundles. Moreover in the work of Beilinson and Drinfeld this global theory for complex algebraic curves (Riemann surfaces) arises from a local-to-global principle, whose local components are algebraic structures underlying conformal field theory (which itself underlies string theory). My proposal, highlighting joint work with D. Nadler, seeks to develop a real version of the geometric Langlands program, describing a harmonic analysis on moduli spaces of real bundles on real algebraic curves, widening the scope of interactions with both classical representation theory and string theory. A strong impetus for this extension is its potential role as an enhancement of the classical representation theory of real semisimple Lie groups. Specifically, we propose that the Langlands classification of representations appears essentially as the special case of our program when the algebraic curve is the projective line. Another motivation arises from the local features of its description on Riemann surfaces with boundary. Here we find a strong interplay with the rapidly emerging physical theory of D-branes, the boundary conditions of conformal field theory.A fundamental theme of modern mathematics is the exploitation of symmetry as an organizing principle, linking diverse and potentially baffling phenomena in an elegant overarching framework. Perhaps the prime example of this trend is the Langlands program, which identifies a general pattern in the appearance of symmetry in number theory, and counts among its successes the solution of Fermat's Last Theorem. In recent years, the Langlands philosophy has been applied in new and more geometric directions, in particular the geometry of surfaces, where it makes contact with the symmetry principles that underly the exciting developments of string theory in physics. My current research proposal (with D. Nadler) suggests an extension of this geometric Langlands program to organize the symmetries associated to surfaces with boundaries or ends. This extension has two main novel attractions. On the one hand, it seeks to encompass, and thereby shed new geometric light on, a classical topic in the study of symmetries involving real numbers. This topic appears naturally in the case when the surface involved is simply a disc. On the other hand, it suggests an intimate new link with one of the most active areas of interest in string theory, the study of the membranes where strings can attach themselves, or end. Thus this proposal provides a new instance of the versatility and unifying appeal of the Langlands philosophy.