arXiv subjects

Maxim Jeffs

Publications and source records attributed to Maxim Jeffs.

4 recordsLinked to original sources

arXiv2026

Fixed Point Floer Cohomology of Dehn Twists I: Splitting Formulas

This paper is the first in a series following on our earlier work [arXiv:2205.14516, arXiv:2307.08180] studying the pair-of-pants product on fixed point Floer cohomology. In [arXiv:2205.14516, arXiv:2307.08180] we fully computed this product for Dehn twists on surfaces of genus greater or equal to 2, and used it to compute a version of the (small) "quantum cohomology" for nodal curves. In the present work, we develop tools for computing the fixed point Floer cohomology and the associated product in the case of Dehn twists in all higher dimensions: for iterated Dehn twists around a Lagrangian sphere in a Liouville domain, we show that the product and differential on the fixed point Floer cohomology split into local and Morse-theoretic contributions on the level of cochains, using some new confinement results for J-holomorphic curves. The local contributions are expected to recover a finite sector of the homology of (twisted) loop spaces of $S^n$ along with an associated Chas-Sullivan product, which we will examine in detail in future work. We also discuss some immediate applications and curiosities for future work.

math.SG↗

arXiv2023

Fixed point Floer cohomology and closed-string mirror symmetry for nodal curves

We show that for singular hypersurfaces, a version of their genus-zero Gromov-Witten theory may be described in terms of a direct limit of fixed point Floer cohomology groups, a construction which is more amenable to computation and easier to define than the technical foundations of the enumerative geometry of more general singular symplectic spaces. As an illustration, we give a direct proof of closed-string mirror symmetry for nodal curves of genus greater than or equal to 2, using calculations of (co)product structures on fixed point Floer homology of Dehn twists due to Yao-Zhao.

math.SG↗

arXiv2022

Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces

We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. These two mirrors are related via a non-geometric equivalence mediated by Kn\"orrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.

math.SG↗

arXiv2020

Mirror symmetry and Fukaya categories of singular hypersurfaces

We consider a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable properties. Firstly, we prove an A-side analog of Orlov's derived Kn\"orrer periodicity theorem by showing that Auroux's category is derived equivalent to the Fukaya-Seidel category of a higher-dimensional Landau-Ginzburg model. Secondly, we describe how this definition implies homological mirror symmetry for some large complex structure limit degenerations of abelian varieties.

math.SG↗