arXiv subjects

Benjamin Gammage

Publications and source records attributed to Benjamin Gammage.

12 recordsLinked to original sources

arXiv2024

Hypertoric Fukaya categories and categories O

To a conical symplectic resolution with Hamiltonian torus action, Braden--Proudfoot--Licata--Webster associate a category O, defined using deformation quantization (DQ) modules. It has long been expected, though not stated precisely in the literature, that category O also admits a "Betti-type" realization as the Fukaya--Seidel category of a Lefschetz fibration. In this paper, we confirm that the category O associated to a toric hyperk\"ahler manifold is equivalent to the partially wrapped Fukaya category of a Liouville manifold stopped by the fiber of a J-holomorphic moment map. The proof involves relating earlier DQ-module computations to a new computation of microlocal perverse sheaves. Leveraging known results on (de Rham) hypertoric category O, we deduce several Floer-theoretic consequences, including formality of simple objects and Koszul duality for the (fully) wrapped Fukaya category; conversely, by applying results about microlocal sheaves, we produce a relative Calabi-Yau structure on category O.

math.SG↗

arXiv2024

Mirror symmetry for Berglund-Hübsch Milnor fibers

We explain how to calculate the Fukaya category of the Milnor fiber of a Berglund-Hübsch invertible polynomial, mostly proving a conjecture of Yankı Lekili and Kazushi Ueda on homological mirror symmetry. As usual, we begin by calculating the "very affine" Fukaya category; afterwards, we deform it, generalizing an earlier calculation of David Nadler. The main step of our calculation may be understood as determining a certain canonical extension of a perverse schober.

math.SG↗

arXiv2023

Hypertoric 2-categories O and symplectic duality

We define 2-categories of microlocal perverse (resp. coherent) sheaves of categories on the skeleton of a hypertoric variety and show that the generators of these 2-categories lift the projectives (resp. simples) in hypertoric category $\mathcal{O}$. We then establish equivalences of 2-categories categorifying the Koszul duality between Gale dual hypertoric categories $\mathcal{O}$. These constructions give a prototype for understanding symplectic duality via the fully extended 3d mirror symmetry conjecture.

math.RT↗

arXiv2023

Perverse schobers and 3d mirror symmetry

The proposed physical duality known as 3d mirror symmetry relates the geometries of dual pairs of holomorphic symplectic stacks. It has served in recent years as a guiding principle for developments in representation theory. However, due to the lack of definitions, thus far only small pieces of the subject have been mathematically accessible. In this paper, we formulate abelian 3d mirror symmetry as an equivalence between a pair of 2-categories constructed from the algebraic and symplectic geometry, respectively, of Gale dual toric cotangent stacks. In the simplest case, our theorem provides a spectral description of the 2-category of spherical functors - i.e., perverse schobers on the affine line with singularities at the origin. We expect that our results can be extended from toric cotangent stacks to hypertoric varieties, which would provide a categorification of previous results on Koszul duality for hypertoric categories $\mathcal{O}$. Our methods also suggest approaches to 2-categorical 3d mirror symmetry for more general classes of spaces of interest in geometric representation theory. Along the way, we establish two results that may be of independent interest: (1) a version of the theory of Smith ideals in the setting of stable $\infty$-categories; and (2) an ambidexterity result for co/limits of presentable enriched $\infty$-categories over $\infty$-groupoids.

math.RT↗

arXiv2022

Betti Tate's thesis and the trace of perverse schobers

We propose a conjecture on the categorical trace of the 2-category of perverse schobers (expected to model the Fukaya-Fueter 2-category of a holomorphic symplectic space). By proving a Betti geometric version of Tate's thesis, and combining it with our previous 3d mirror symmetry equivalence and the Ben-Zvi--Nadler--Preygel result on spectral traces, we are able to establish our conjecture in the simplest interesting case.

math.RT↗

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↗

arXiv2022

Homological mirror symmetry at large volume

A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove homological mirror symmetry: the category of coherent sheaves on the first space is equivalent to the Fukaya category of the second. Our equivalence intertwines the Viterbo restriction maps for a generalized pair-of-pants cover of the symplectic manifold with the restriction of coherent sheaves for a certain affine cover of the algebraic variety. We deduce a posteriori a local-to-global principle conjectured by Seidel -- certain diagrams of Viterbo restrictions are cartesian -- by passing Zariski descent through our mirror symmetry result.

math.SG↗

arXiv2021

Local mirror symmetry via SYZ

In this note, we explain how mirror symmetry for basic local models in the Gross-Siebert program can be understood through the non-toric blowup construction described by Gross-Hacking-Keel. This is part of a program to understand the symplectic geometry of affine cluster varieties through their SYZ fibrations.

math.SG↗

arXiv2021

Mirror Symmetry for Truncated Cluster Varieties

In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cluster variety, we construct a symplectic manifold and prove homological mirror symmetry for the resulting pair. We also describe how our construction can be obtained from toric geometry, and we relate our construction to various aspects of cluster theory which are known to symplectic geometers.

math.SG↗

arXiv2021

Mirror symmetry for very affine hypersurfaces

We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using recent results on localization, we may trade wrapped Fukaya categories for microlocal sheaf theory along the skeleton of the hypersurface. Using Mikhalkin-Viro patchworking, we identify the skeleton of the hypersurface with the boundary of the Fang-Liu-Treumann-Zaslow skeleton. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties.

math.SG↗

arXiv2019

Mirror symmetry for honeycombs

We prove a homological mirror symmetry equivalence between an $A$-brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a $B$-brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the $A$-model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces.

math.SG↗

arXiv2019

Homological mirror symmetry for hypertoric varieties II (with an Appendix written jointly with Laurent C\^ot\'e and Justin Hilburn)

In this paper, we prove a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties, and we calculate monodromy autoequivalences of these categories by promoting our result to an equivalence of perverse schobers. We prove our equivalence by matching holomorphic Lagrangian skeleta, on the A-model side, with non-commutative resolutions on the B-model side. The hyperk\"ahler geometry of these spaces provides each category with a natural t-structure, which helps clarify SYZ duality in a hyperk\"ahler context. Our results are a prototype for mirror symmetry statements relating pairs of K-theoretic Coulomb branches.

math.AG↗