Open Mirror Symmetry for Toric Varieties

Columbia University, New York NY

Investigators

Abstract

Mirror symmetry as a duality from string physics builds a bridge between seemingly very different mathematical theories on one manifold and its mirror manifold. Open mirror symmetry is this duality in the open string sector, which involves mathematical theories especially associated to open strings. This project studies open mirror symmetry from both enumerative and categorical aspects. In the enumerative aspect, the PI proposes to investigate and prove the enumerative prediction (remodeling conjecture) from mirror symmetry for toric Calabi-Yaus in a multi-step project, including the cases of toric orbifolds. The remodeling conjecture uses a recursive relation to predict higher genus topological string amplitudes. When applied to orbifolds, this conjecture predicts Gromov-Witten potential currently unknown by other methods, such as topological vertex. As for the categorical aspect, this project continues the PI and collaborators' previous work on Kontsevich's homological mirror symmetry (HMS), especially around T-duality and coherent-constructible correspondence (CCC) that PI et. al. construct. In particular, the PI plans to construct a geometric T-duality procedure for arbitrary vector bundles on toric varieties and toric normal crossings, and prove this procedure is consistent with the CCC, which is combinatorial in nature. In the interplay of these two aspects, the calculation of disk invariants is an essential ingredient in explicitly studying HMS with obstructed Lagragians, such as HMS for toric Calabi-Yau threefolds, which this project also plans to address. String theory is a candidate physics theory to describe all known fundamental forces. It requires sophisticated mathematics, and vice-versa, string theory has a long-lasting and deep influence on modern mathematics. There are several seemingly different set-ups of string theories ? if string theory wants to be the theory of this universe, there cannot be several different theories ? fortunately furthermore there are string dualities relating them. It turns out that these different set-ups are equivalent, and there is essentially a single string theory. Mirror symmetry is one of these dualities that equates two types of set-ups (IIA and IIB). In mathematics, this translates into beautiful and unforeseen connections between two vastly distinct areas of geometry. This project studies several mathematical aspects of mirror symmetry among toric varieties, a kind of spaces that serve as a simplified model of the extra-dimension in string physics. It contributes to the basic understanding of relevant mathematics, such as algebraic geometry, and also provides a concrete approach to a more complicated string theory from an equivalent-but-easier way.