arXiv research

The paper studies the geometry of Nakajima quiver varieties and their decompositions.

Study of conformal limits in Nakajima quiver varieties.

Introduces Nakajima bundles on algebraic curves, generalizing quiver representations and bundles.

Classifies singularities in quiver varieties for specific Dynkin quivers.

Quiver varieties' geometry at infinity studied using Nakajima metric.

The paper trivializes moment maps for various geometric structures.

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne-Simpson…

The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.

Paper develops Morse-theoretic approach to quiver varieties convolution.

Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.

Generalizing work of Haydys and Hitchin, we prove the existence of a hyperholomorphic line bundle on certain hyperkähler manifolds that do not necessarily admit an $S^1$ action. As examples, we consider the moduli space of (non-strongly) parabolic Higgs bundles, the moduli space of solutions to Nahm's equations, and Na…

We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such da…

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawren…

Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.

The paper connects hyperpolygon spaces to Higgs bundle moduli spaces via degenerations.

We show that on Hilbert scheme of $n$ points on $\C^2$, the hyperkähler metric construsted by H. Nakajima via hyperkähler reduction is the Quasi-Asymptotically Locally Euclidean (QALE in short) metric constructed by D. Joyce.

Study of Hilbert schemes and Coulomb branches of hypertoric varieties.

In this paper we investigate the relation between complexified Fenchel-Nielsen coordinates and spectral network coordinates on Seiberg-Witten moduli space. The main technique is the comparison of exact expressions for the expectation value of 't Hooft defects in certain 4D $SU(2)$ $\mathcal{N}=2$ gauge theories. We der…

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many…

Some moduli spaces of irregular connections on the trivial bundle over the Riemann sphere will be identified with Nakajima quiver varieties. In particular this enables us to associate a Kac-Moody root system to such connections (yielding many isomorphisms between such moduli spaces, via the reflection functors for the …

Monoidal categorifies genus zero skein algebra using K-theory.

This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannia…

Let $S$ be a smooth rational curve on a complex manifold $M$. It is called ample if its normal bundle is positive. We assume that $M$ is covered by smooth holomorphic deformations of $S$. The basic example of such a manifold is a twistor space of a hyperkahler or a 4-dimensional anti-selfdual Riemannian manifold $X$ (n…

In this thesis we study the topology and geometry of hyperkähler quotients, as well as some related non-compact Kähler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theo…

Constructs moduli stacks of quiver bundles and applies to Higgs bundles.

We prove a weighted Sobolev inequality and a Hardy inequality on manifolds with nonnegative Ricci curvature satisfying an inverse doubling volume condition. It enables us to obtain rigidity results for Ricci flat manifolds, generalizing earlier work of Bando, Kasue and Nakajima.

New techniques compute $L^2$-cohomology of quasi-fibered metrics.

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebra…

In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in $\R^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.

We consider a hyperkähler reduction and describe it via frame bundles. Tracing the connection through the various reductions, we recover the results of Gocho and Nakajima. In addition, we show that the fibers of such a reduction are necessarily totally geodesic. As an independent result, we describe O'Neill's submersio…

Classifies instantons on ALF multi-Taub-NUT spaces and ties them to bow solutions.

The Kapustin-Witten equations relate to nonabelian Hodge theory on Kähler surfaces.

We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between $K$-theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshiok…

We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/Γ$, where $M$ is a smooth manifold and $S^3/Γ$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associ…

Quantized Coulomb branches linked to skein algebras.

Proves unique ALE instanton with toric Hermitian structure.

In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical poin…

For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function of $K$. Let $K, K_0\subset \R^n$ be smooth convex bodies of class $C^2_+$, and l…

Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-v…

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid H…

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kähler t…

We conjecture a formula for the generating function of virtual $χ_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. The…

A result from Gromov ensures the existence of a contact structure on any connected non-compact odd dimensional Lie group. But in general such structures are not invariant under left translations of the Lie group. The problem of finding which Lie groups admit a left invariant contact structure (contact Lie groups), is t…

In this paper we completely classify symplectic actions of a torus $T$ on a compact connected symplectic manifold $(M, σ)$ when some, hence every, principal orbit is a coisotropic submanifold of $(M, σ)$. That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action …

In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to rem…

We prove the ADO invariants are a q-holonomic family and establish recursion relations.

Introduces a new geometric structure for statistical manifolds with degenerate metrics.

Formula derived for mass of almost Kähler manifolds, extending previous results.