A Canonical Construction of Mirrors for Polarized Calabi-Yau Manifolds

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This award supports research in algebraic geometry. Focused primarily on varieties defined by polynomial equations, algebraic geometry is an ancient subject that plays a key role in numerous fields of mathematics, both pure and applied, as well as in physics. The main theme of this project is that a broad class of geometric objects, so called Calabi-Yau varieties, come with a natural system of coordinates. Informally speaking, creatures living on a Calabi-Yau variety should be able to perceive natural, intrinsic quantities whose values determine their precise position. As these geometric objects play a fundamental role in diverse areas of mathematics, these intrinsic quantities should play a similar fundamental role in theoretical physics, particularly in superstring theory. Since string theory models suggest that extra dimensions of spacetime comprise a Calabi-Yau variety, this study suggests there are corresponding fundamental intrinsic quantities, not yet understood, in our world. This research project aims to deepen and advance knowledge in this field.The main objective of the proposed research is to continue the investigator's collaborative work to generalize the classical theory of theta functions for abelian varieties to polarized Calabi-Yau varieties, both open (i.e. log) and compact. More precisely, the goal is to give a canonical basis for the vector space of global sections, and a formula for the structure constants for the multiplication rule in the coordinate ring, expressed in the canonical basis, as counts of holomorphic disks on the mirror. The existence of such generalized theta functions points to the existence of a geometrically meaningful compactification of the moduli space, vastly generalizing the compactificaton of polarized abelian varieties and the theory of the secondary polytope, and at the same time suggests a synthetic construction of the mirror in terms of the canonically described ring. The project includes a detailed scheme for constructing the compactification in dimension two, and the synthetic construction of the mirror in full generality, by counting rigid analytic disks.