The Hamiltonian reduction of hypertoric mirror symmetry
The wrapped Fukaya category of a multiplicative hypertoric variety was described in my paper Homological mirror symmetry for hypertoric varieties II, with Ben Gammage and Ben Webster. In this paper, we consider the natural torus action on this category, and explain how to define its symplectic reduction at singular values of the moment map parameter. We obtain in this way the Fukaya category of a toric hyperplane arrangement.
Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers
with Roman Bezrukavnikov, Pablo Boixeda Alvarez and Zhiwei Yun. Pure and Applied Mathematics Quarterly (Special volume for G.Lusztig) pp. 61-130 Volume 21 (2025) Number 1
We construct moduli spaces of higgs bundles, flat connections and local systems on the line with irregular singularities at infinity, which are to homogeneous affine springer fibers as Slodowy slices are to usual Springer fibers.
Elliptic stable envelopes and hypertoric loop spaces.
with Artan Sheshmani and Shing-Tung Yau. Selecta Mathematica 29.5 (2023): 73.
This paper relates the elliptic stable envelopes of a hypertoric variety with the K-theoretic stable envelopes of the loop hypertoric space. It thus points to a possible categorification of elliptic stable envelopes.
Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces
with Artan Sheshmani and Shing-Tung Yau. Annales de l'Institut Fourier (forthcoming).
We study moduli spaces of twisted quasimaps to a hypertoric variety X, arising as the Higgs branch of an abelian supersymmetric gauge theory in three dimensions. These parametrise general quiver representations whose building blocks are maps between rank one sheaves on P1, subject to a stability condition, associated to the quiver, involving both the sheaves and the maps. We show that the singular cohomology of these moduli spaces is naturally identified with the Ext group of a pair of holonomic modules over the `quantized loop space' of X, which we view as a Higgs branch for a related theory with infinitely many matter fields. We construct the coulomb branch of this theory, and find that it is a periodic analogue of the coulomb branch associated to X. Using the formalism of symplectic duality, we derive an expression for the generating function of twisted quasimap invariants in terms of the character of a certain tilting module on the periodic coulomb branch. We give a closed formula for this generating function when X arises as the abelianisation of the N-step flag quiver.
Hypertoric Hitchin systems and Kirchhoff polynomials
with Michael Groechenig. International Mathematics Research Notices 2022.19 (2022): 15271-15312.
We define a formal algebraic analogue of hypertoric Hitchin systems, whose complex-analytic counterparts were defined by Hausel-Proudfoot. These are algebraic completely integrable systems associated to a graph. We study the variation of the Tamagawa number of the resulting family of abelian varieties, and show that it is described by the Kirchhoff polynomial of the graph. In particular, this allows us to compute their p-adic volumes. We conclude by remarking that these spaces admit a volume preserving tropicalisation.
Deletion-Contraction Triangles for Hausel-Proudfoot varieties
with Zsuzsanna Dancso and Vivek Shende. Journal of the European Mathematical Society (EMS Publishing) 26.7 (2024).
To any graph, Hausel and Proudfoot associate a "Betti"and a "Dolbeault" space, which behave respectively like the moduli of local systems and a moduli of higgs bundles. The former is affine, and is in fact a multiplicative hypertoric variety. The latter carries a complex integrable system approximating the Hitchin fibration. We produce a diffeomorphism between these spaces, and show that it intertwines the weight filtration on the cohomology of the Betti space with the perverse-Leray filtration on the Dolbeault space. In the process, we construct triangles relating the Betti and Dolbeault spaces for a graph, its contraction along an edge its deletion along the same edge.
Homological mirror symmetry for hypertoric varieties II
with Ben Gammage and Ben Webster. Geometry and Topology (forthcoming).
We show that the mirror, in the sense of homological mirror symmetry, of a multiplicative hypertoric variety is another multiplicative hypertoric variety. As with the first paper in this series, we avoid working with the Fukaya category directly. In this case, rather than deformation quantization modules, we use microlocal sheaves along the Liouville skeleton of the hypertoric variety, leaning on recent work of Ganatra, Pardon and Shende.
The quantum Hikita conjecture
with Joel Kamnitzer and Nick Proudfoot. Advances in Mathematics 390 (2021): 107947.
Symplectic resolutions tend to come in symplectically dual pairs, which appear as the Higgs and Coulomb branches of three-dimensional gauge theories. Such pairs are expected to satisfy a number of remarkable relations: here we add to the list, by conjecturing an equality of the quantum D-module (in the Calabi-Yau specialization) of a symplectic resolution with the "character D-module" of its dual.
Homological mirror symmetry for hypertoric varieties I
with Ben Webster. Geometry & Topology 28.3 (2024): 1005-1063.
We establish a dictionary between the modular representation theory of an additive hypertoric variety and a category of characteristic zero deformation quantization modules on a multiplicative hypertoric variety. We explain how this factors through a form of homological mirror symmetry, where DQ modules play the role of the Fukaya category. Of special interest is the presence of a dilation action on the additive variety, which becomes a Hodge grading on the mirror.
Intersection cohomology and quantum cohomology of conical symplectic resolutions
with Nick Proudfoot. Algebraic Geometry 2 (2015), no. 5, 623--641.
In all cases computed so far, the quantum cohomology ring of a conical symplectic resolution is defined on a Zariski open subset of the Kähler parameters. Along the complement, the structure coefficients acquire poles. We extend the ring to the complement of this subset, and specialise to the most singular possible value. We prove in the hypertoric case, and conjecture in general, that this specialisation equals the intersection cohomology of the affinized symplectic resolution. In particular, this endows the intersection cohomology with a ring structure.
Quantum cohomology of hypertoric varieties
with Daniel K. Shenfeld. Letters in Mathematical Physics 103 (2013), no. 11, 1273--1291.
We compute the equivariant quantum cohomology ring of a smooth hypertoric variety. In particular, we find a presentation by generators and relations, rather similar to the famous Bethe equations arising in mathematical physics, and find a full set of integral solutions to the quantum differential equation. This last result, in the spirit of homological mirror symmetry, suggests that the mirror is a multiplicative hypertoric variety, an idea which is explored in my work with Ben Webster and Ben Gammage above.
Quantum cohomology of hypertoric varieties and geometric representations of Yangians
This was my doctoral thesis, under the supervision of Andrei Okounkov. In the first part, I show that the Maulik-Okounkov Yangian, acting on the cohomology of a Nakajima quiver variety of type ADE, coincides with the action defined by Varagnolo. The second half recapitulates my paper with Daniel Shenfeld, above.