Calabi-Yau Varieties: Arithmetic, Geometry and Physics

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The proposed program is concerned with problems at the crossroads of number theory (arithmetic algebraic geometry) and theoretical physics (string theory). The outcomes will enhance our understanding in the chosen fields of mathematics and physics, and ultimately of our universe. String theory is a physics prediction that what the universe is made of, is, at the most elementary level, strings (one-dimensional objects), rather than point particles (zero-dimensional objects). String theory demands ten dimensional space-time (as opposed to the dimension four of our real world). This is, roughly speaking, because more dimensions can accommodate more possible string vibrations. The extra six dimensional objects in string theory are known as Calabi-Yau manifolds (threefolds), and they are the main concern in my investigation. The objective of the proposed research is to understand the physical prediction of mirror symmetry for Calabi-Yau manifolds from a mathematical point of view. A Calabi-Yau manifold is a compact complex Kaehler manifold with vanishing first Chern class and zero first Betti number. Calabi-Yau manifolds of dimension one, two, and three are, respectively, elliptic curves, K3 surfaces, and Calabi-Yau threefolds. Mirror symmetry is a prediction in string theory that certain “mirror pairs” of Calabi-Yau threefolds yield identical physical theories. Modular (and quasimodular) forms, Hilbert, Siegel and Jacobi modular forms, and automorphic forms appear, as generating functions of geometric invariants, or as partition functions, or as mirror maps in the mirror symmetry landscape. One of my goals is to interpret mirror symmetry in terms of arithmetic invariants such as zeta-functions and L-series of the Calabi-Yau manifolds in question. In this regard, the automorphy question for the L-series will be vigorously pursued. Here, “automorphy” refers to the presumed fact that the (motivic) L-series arising from Calabi-Yau manifolds are automorphic L-series in the context of the Langlands program. The most compelling endeavour in this project is to consider the above problem for families of Calabi-Yau manifolds with deformation parameters. Such families arise when we consider mirror Calabi-Yau manifolds. We ought to introduce automorphic (modular) forms with parameters. Another central goal of my project is the conceptual understanding of the modular properties of the partition functions, and of the generating functions of the Gromov-Witten invariants, the Donaldson-Thomas invariants and other geometric invariants, such as instanton number, for Calabi-Yau manifolds. This will involve studying, among other things, Feynman path integrals on trivalent graphs, the inductive nature of such graphs, and their interpretation in terms of Kodaira-Spencer theory and Bershadsky-Cecotti-Ooguri-Vafa (BCOV) theory. It is imperative to lay solid mathematical foundations for string theory, for the benefit of both mathematicians and string theorists.