Mirror Symmetry and Moduli of Calabi-Yau Manifolds

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

String theory asserts that our universe has ten dimensions: the usual three spatial dimensions and time, together with six microscopic dimensions which form a geometric object at each point called a Calabi--Yau manifold. The geometry of the Calabi--Yau manifold encodes the interactions of subatomic particles in a way that is compatible with the theory of gravity. There are millions of different types of Calabi--Yau manifolds known, but it is conjectured by Miles Reid that they are all related by simple "surgery" operations. The physical theory changes in a continuous way under these surgeries, so Reid's conjecture implies that the physical theories associated to different Calabi--Yau manifolds are all connected. The mirror symmetry phenomenon posits that Calabi--Yau manifolds come in pairs X and Y which determine equivalent physical theories. leading to an intricate but mysterious relation between the geometries of X and Y. In this project, the PI will study the mirror symmetry phenomenon and its relation to Reid's conjecture and the classification of Calabi--Yau manifolds. The project also provides research training opportunities for graduate students. The mirror symmetry phenomenon of string theory implies that Calabi--Yau manifolds come in pairs X and Y such that the complex geometry of X is related to the symplectic geometry of Y, and vice versa. More precisely, the homological mirror symmetry (HMS) conjecture of Kontsevich asserts that the derived category of holomorphic vector bundles on X is equivalent to the Fukaya category of Lagrangian submanifolds of Y. The related Strominger--Yau--Zaslow (SYZ) conjecture asserts that X and Y admit dual Lagrangian torus fibrations over a common base, and the HMS equivalence is obtained via an analogue of the Fourier transform. The PI and a collaborator will establish new cases of the HMS conjecture for Calabi--Yau manifolds of complex dimensions 2 and 3, guided by the SYZ heuristic. The PI will use mirror symmetry to study moduli spaces of Calabi--Yau 3-folds and Reid's conjecture, including the case of non-Kahler Calabi--Yau 3-folds which plays a key role in the conjecture. The PI will pursue several related projects with undergraduate students, graduate students, and postdocs, as follows: mirror symmetry for smoothings of triangle singularities, classification of Calabi--Yau 3-folds fibered in abelian surfaces, HMS for non-compact analogues of Enriques surfaces, HMS for Fano varieties and their Kuznetsov components, and Kollar's conjecture on deformations of rational surface singularities. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.