Related papers: Hypertoric varieties, $W$-Hilbert schemes, and Cou…

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…

In this thesis we study the topology and geometry of hyperk\"ahler quotients, as well as some related non-compact K\"ahler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory…

We study variants of Hikita conjecture for Nakajima quiver varieties and corresponding Coulomb branches. First, we derive the equivariant version of the conjecture from the non-equivariant one for a set of gauge theories. Second, we suggest…

We develop the diagrammatic technique of quiver subtraction to facilitate the identification and evaluation of the $\mathrm{SU}(n)$ hyper-K\"ahler quotient (HKQ) of the Coulomb branch of a $3d$ $\mathcal{N}=4$ unitary quiver theory. The…

Equivariant localization expresses global invariants in terms of local invariants, and many of them appearing in equivariant index theory, (holomorphic) Morse theory, geometric quantization and supersymmetric localization can be…

Hypertoric varieties are hyperk\"ahler analogues of toric varieties, and are constructed as abelian hyperk\"ahler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold…

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their anti-symplectic cousins, which are compositions of a diagram automorphism, a reflection functor and a transpose defined by certain bilinear forms. These…

For algebraic varieties defined by hyperkahler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the "hyperkahler Kirwan map" on cohomology is surjective. We resolve this question in the affirmative…

Braverman, Finkelberg and Nakajima have recently given a mathematical construction of the Coulomb branches of a large class of $3d$ $\mathcal{N} =4$ gauge theories, as algebraic varieties with Poisson structure. They conjecture that these…

We show that Coulomb branches for $3$-dimensional $\mathcal{N}=4$ supersymmetric gauge theories have symplectic singularities. This confirms a conjecture of Braverman-Finkelberg-Nakajima.

We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical…

We show that scale-invariant special Kahler geometries whose generic r-complex-dimensional abelian variety fiber is isomorphic (completely split) or isogenous (completely iso-split) as a complex torus to the product of r one-dimensional…

When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a…

We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions,…

We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke (Inventiones 1988) on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincare upper half plane,…

Using the graphical method developed in hep-th/9908082, we obtain the full curve corresponding to the hyperk\"ahler quotient from the extended E_7 Dynkin diagram. As in the E_6 case discussed in the same paper above, the resulting curve is…

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…

We study the Coulomb-branch sector of 3D $\mathcal{N}=4$ gauge theories with half-hypermultiplets in general pseudoreal representations $\mathbf{R}$ ("noncotangent" theories). This yields (short) quantization of the Coulomb branch and…

We show that the Looijenga--Lunts--Verbitsky Lie algebra acting on the cohomology of a hyperk\"ahler variety is a derived invariant, and obtain from this a number of consequences for the action on cohomology of derived equivalences between…

In 2009 Gaiotto, Moore and Neitzke presented a new construction of hyperk\"{a}hler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration $\mathcal{M}$ over a base space $\mathcal{B}$, except for…