This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This proposal explores several areas where algebraic geometry interacts with quantum field theory and string theory: the Geometric Langlands Program, on the math side; heterotic string phenomenology and F theory, in physics; and the superstring measure, an algebraic geometry project motivated by physics. The recent breakthrough in producing a Heterotic Standard Model is a perfect illustration of the power of algebraic geometry at the service of physics. Using techniques for construction of non simply connected Calabi-Yau threefolds and of bundles on them satisfying various constraints on their chern classes and cohomology, the PI produced the only known example of a heterotic string compactification which has exactly the Minimal Supersymmetric Standard Model (MSSM) spectrum of particles and forces, with no unwanted exotic matter. A systematic study is proposed of the High Country region of the string Landscape, where the Heterotic Standard Models live. This includes investigation of all known non simply connected Calabi-Yau threefolds, incorporating a classification of all Standard Model bundles on them and analysis of their mathematical and phenomenological properties. The apparent great scarcity of these Heterotic Standard Models motivates attempts to determine the rough size of the string High Country. The recent phenomenological breakthroughs based on F-theory underlie the urgency of constructing global, geometric models realizing the various known local models. The construction, at all genera, of the superstring measure is an important foundational issue in string theory. Recent proposals have converted this to a question in classical algebraic geometry, closely related to modular forms, the Schottky problem, and theta identities. The existing proposals do not quite work. Fortunately, it seems likely that the addition of some algebro-geometric ingredients may overcome the obstruction.
The Geometric Langlands Conjecture is an old and central open problem in algebraic geometry and representation theory. In recent years it has also been of great interest to physicists, who have embedded it into the context of quantum field theory. The combination of their physical insights with recent breakthroughs in non abelian Hodge theory and older ideas from integrable systems offers the real possibility of a complete solution soon. F-theory and its duality to the heterotic string are another area where algebraic geometry is able to make powerful contributions to the physics. The PI also proposes to continue a wide range of educational activities, including curriculum development, the writing of a textbook, and extensive work with undergraduate and graduate students, aimed at the dissemination of new knowledge concerning the interactions of mathematics and high energy physics.
