Algebraic Geometry and String Theory

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Donagi 9802456 This project attempts to explore some of the deep and beautiful interrelationships discovered and developed recently between algebraic geometry and high energy physics. The PI will study three issues, each involving a combination of ideas from algebraic geometry, algebraically integrable systems, quantum field theory and string theory. Of these, the first is concerned with applications of an algebro-geometric idea to a string duality conjecture, suggesting along the way a new description of an algebro-geometric moduli space; the second explores a construction of integrable systems which is motivated by stringy ideas and which is expected in turn to lead to new descriptions for certain supersymmetric quantum field theories; and the third investigates a problem within algebraic geometry in light of recent insights from string theory. (1) Principal bundles on elliptic fibrations, spectral covers, and Heterotic/F-theory duality. It is proposed to apply and adapt previous results on spectral covers, obtained in the context of integrable systems, to the description of the moduli space of principal bundles on an elliptically fibered variety, and to the comparison of both the moduli spaces and the superpotentials arising from compactifications of the Heterotic string with those coming from F-theory. (2) Calabi-Yau vs. Seiberg-Witten integrable systems. Ideas from string theory suggest a construction and classification of Seiberg-Witten integrable systems in terms of degenerations of the Calabi-Yau system constructed by Donagi and Markman. It is hoped that this in turn will allow the construction of the missing SW systems, including those of the theory with adjoint matter for an arbitrary gauge group. (3) The Geometric Langlands Conjecture. A partial construction due to the PI, based on "classical" integrable systems, will be compared in detail to another, based on a quantized analogue, and the possibility will be explored that a certain "stringy" system provides the correct common extension of both constructions, producing all the automorphic sheaves required by the geometric Langlands conjecture. This is research on the boundary of algebraic geometry and string theory. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. String theory is the newest and most exciting candidate for a physical theory unifying the four fundamental forces of nature. In the last few years, the two fields have interacted at great depth and across a broad frontier of problems of common interest. These interactions have led to some of the most exciting recent breakthroughs in both fields, and hold the promise of leading to a physical "theory of everything" as well as to an algebraic geometry which fully incorporates quantum and stringy phenomena.