2025 Theses Doctoral

Open Enumerative Mirror Symmetry for Lines in the Mirror Quintic

Haney, Ryuichi Sebastian

Mirror symmetry is a family of conjectural equivalences between the symplectic geometry (𝐴-model) of a symplectic manifold and the algebraic geometry (𝐵-model) of a corresponding mirror space. The earliest mathematical achievement of mirror symmetry was a prediction for the genus zero Gromov--Witten invariants of the quintic threefold in terms of period integrals on a mirror family of Calabi--Yau varieties.

Using homological mirror symmetry, we deduce an analogous mirror theorem for the open Gromov--Witten invariants of a particular Lagrangian submanifold in the quintic threefold assuming the existence of a negative cyclic open-closed map. The Lagrangian we consider can be thought of as an SYZ mirror to a line in the mirror quintic and their open Gromov-Witten (OGW) invariants coincide with the Abel--Jacobi images of these lines as calculated by Walcher.

In this example, the OGW invariants are irrational numbers contained in ℚ(√-3) and admit an expression involving special values of a Dirichlet L-function. The field in which the OGW invariants lie arises as the invariant trace field of (the smooth locus of) a closely related hyperbolic Lagrangian submanifold with conical singularities in the quintic. These results explain some of the predictions on the existence of hyperbolic Lagrangian submanifolds in the quintic put forward by Jockers, Morrison, and Walcher. We achieve these results by computing the Lagrangian Floer theory of a different Lagrangian immersion in the quintic supporting a family of objects in the Fukaya category that are homologically mirror to coherent sheaves supported on line in the mirror quintic.

Some of the key technical ingredients in this proof include a definition of the wrapped Fukaya category with Lagrangian immersions, an adjunction isomorphism on wrapped Floer cohomology associated to a non-compact Lagrangian immersion, a construction of the open Gromov--Witten invariants from a strong proper Calabi--Yau structure on the Fukaya category, and a framework for relating open Gromov--Witten invariants to Abel--Jacobi images under homological mirror symmetry.