arXiv subjects

Sebastian Haney

Publications and source records attributed to Sebastian Haney.

4 recordsLinked to original sources

arXiv2024

Open enumerative mirror symmetry for lines in the mirror quintic

Mirror symmetry gives predictions for the genus zero Gromov-Witten invariants of a closed Calabi--Yau variety in terms of period integrals on a mirror family of Calabi-Yau varieties. We deduce an analogous mirror theorem for the open Gromov-Witten invariants of certain Lagrangian submanifolds of the quintic threefold from homological mirror symmetry, assuming the existence of a negative cyclic open-closed map. The Lagrangians we consider can be thought of as SYZ mirrors to lines, and their open Gromov-Witten (OGW) invariants coincide with relative period integrals on the mirror quintic calculated by Walcher. Their OGW invariants are irrational numbers contained in an algebraic extension of the rationals, and admit an expression similar to the Ooguri-Vafa multiple cover formula involving special values of a Dirichlet L-function. We achieve these results by studying the Floer theory of a different immersed Lagrangian in the quintic that supports a one-dimensional family of objects in the Fukaya category homologically mirror to coherent sheaves supported on lines in the mirror quintic. 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-Walcher.

math.SG↗

arXiv2024

Infinity inner products and open Gromov--Witten invariants

The open Gromov--Witten (OGW) potential is a function from the set of weak bounding cochains on a closed Lagrangian in a closed symplectic manifold to the Novikov ring. Existing definitions of the OGW potential assume that the ground field of the Novikov ring is either $\mathbb{R}$ or $\mathbb{C}$. In this paper, we give an alternate definition of the OGW potential in the pearly model for Lagrangian Floer theory which yields an invariant valued in the Novikov ring over any field of characteristic zero. We work under simplifying regularity hypotheses which are satisfied, for instance, by any monotone Lagrangian. Our OGW potential is defined in terms of an appropriate weakening of a strictly cyclic pairing on a curved $A_{\infty}$-algebra, which can be thought of as a version of a proper Calabi--Yau structure. Such a structure is obtained by constructing a version of the cyclic open-closed map on the pearly Lagrangian Floer cochain complex. We also explain an analogue of our construction in de Rham cohomology, and show that it recovers the OGW potential constructed by Solomon and Tukachinsky.

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arXiv2024

Cusped hyperbolic Lagrangians as mirrors to lines in three-space

We construct a Lagrangian in the cotangent bundle of a 3-torus whose projection to the fiber is a neighborhood of a tropical curve with a single 4-valent vertex. This Lagrangian has an isolated conical singular point, and its smooth locus is diffeomorphic to the minimally-twisted five component chain link complement, a cusped hyperbolic 3-manifold. From this singular Lagrangian, we construct an immersed Lagrangian, and determine when it is unobstructed in the wrapped Fukaya category. We show that for a generic line in projective 3-space, there is a local system on this immersed Lagrangian such that the resulting object of the wrapped Fukaya category is homologically mirror to an object of the derived category supported on the line. In the course of the proof, we construct a version of the wrapped Fukaya category with objects supported on Lagrangian immersions, which may be of independent interest.

math.SG↗

arXiv2019

Cylindrical contact homology of 3-dimensional Brieskorn manifolds

Cylindrical contact homology is a comparatively simple incarnation of symplectic field theory whose existence and invariance under suitable hypotheses was recently established by Hutchings and Nelson. We study this invariant for a general Brieskorn 3-manifold $\Sigma(a_1,\ldots, a_n)$, and give a complete description of the cylindrical contact homology for this 3-manifold equipped with its natural contact structure, for any $a_j$ satisfying $\frac{1}{a_1} + \cdots + \frac{1}{a_n} < n-2$.

math.SG↗