Algebraic Cycles, Hodge Theory, and Arithmetic

Washington University, Saint Louis MO

Investigators

Abstract

The last two decades have seen a highly productive interaction between Hodge theory, symplectic geometry, and string theory in the mirror symmetry program. Recent discoveries suggest a new cross-fertilization between theoretical physics and algebraic K-theory, deepening the existing connection with topological K-theory. To make this new connection explicit, the P.I. proposes to construct families of cycles in low-degree algebraic K-groups of Calabi-Yau varieties, and to study their behavior under degeneration and homological mirror symmetry. This part of the project will have applications to the arithmetic of Gromov-Witten invariants. Underlying Hodge-theoretic problems which will be addressed include the classification of period subdomains parametrizing Calabi-Yau threefolds, and the determination of their Kato-Usui boundary components and automorphic cohomology. These results will be relevant not only to Hodge theory and physics but also for automorphic forms and the Langlands program. They will, in addition, involve the construction of new Calabi-Yau varieties. Hodge theory seeks to describe the influence of integrals and differential equations on the shape of an algebraic space. Famous conjectures including that of Hodge (which is one of the seven Clay Millenium Problems) predict that these computations, while non-algebraic a priori, are firmly governed by structures called algebraic groups and algebraic cycles. But these latter structures are far more ubiquitous, and useful, than the conjectures suggest; the P.I. will investigate their application to related problems in number theory and physics. For example, in the model of spacetime proposed by string theory (which purports to unify quantum mechanics and general relativity), 6 as-of-yet unobserved real dimensions are accounted for by Calabi-Yau spaces. These spaces are linked by dualities which preserve the physical theory while completely altering the mathematics. Incorporating algebraic cycles and their generalizations into these dualities will completely explain observed asymptotics of instanton numbers. This is part of the thrust of an interdisciplinary BIRS workshop on "Hodge Theory and String Duality" (Dec. 2011) co-organized by the P.I. Results from this project will be disseminated through such conferences, summer schools, journal articles and websites. The project consultants brought to Washington University by the grant will contribute to the research atmosphere, and there are specialized problems related to the project which will be suitable for training graduate students.