Theta Functions for Polarized Calabi-Yau Varieties

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The main objective of the proposed research is 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: 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, determined by counts of rational curves on the mirror. The existence of such generalized theta functions points to the existence of geometrically meaningful compactification of the moduli space, vastly generalizing the Mumford-Namikawa-Alexeev-Olsson compactificaton of A_{g,d}, and the Gelfand-Kapranov-Zelevinski-Alexeev-Olsson theory of the secondary polytope, and at the same time suggests a synthetic construction of the mirror as Proj of the canonically described ring. The proposal includes a detailed scheme for carrying this out in dimension two, and for cluster varieties of all dimensions.The main proposal is that a broad class of geometric objects, so called Calabi-Yau varieties, come with a natural system of coordinates. Informally: If you live on a Calabi-Yau variety, there should be natural, intrinsic quantities in your world, whose values determine your precise position. As these geometric objects play a fundamental role in diverse areas of mathematics, these intrinsic quantities should play a similar fundamental role. More broadly, string theory models suggest that WE live on a Calabi-Yau, and thus the proposal suggests there are such fundamental quantities, not yet understood, in our world.