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Hiraku Nakajima

Publications and source records attributed to Hiraku Nakajima.

At least 37 records · Page 2Linked to original sources
arXiv2016

Lectures on perverse sheaves on instanton moduli spaces

I will explain my joint paper `Instantons moduli spaces and W-algebras' with A.Braverman, M.Finkelberg, arXiv:1406.2381. I will concentrate on the geometric part, that is a study of perverse sheaves on instanton moduli spaces. I place a particular emphasize on the hyperbolic restriction functor and stable envelop, which are our key tools, and appear also in other situations in geometric representation theory. These are lecture notes for a course in 2015 PCMI.

math.RT↗
arXiv2016

Instanton moduli spaces and $\mathscr W$-algebras

We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincaré pairing) in terms of representation theory of some vertex operator algebras ("$\mathscr W$-algebras").

math.QA↗
arXiv2016

Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, I

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold, such as instanton moduli spaces on $\mathbb R^4$, $SU(2)$-monopole moduli spaces on $\mathbb R^3$, etc. In this paper and its sequel, we propose a mathematical definition of the coordinate ring of the Coulomb branch, using the vanishing cycle cohomology group of a certain moduli space for a gauged $σ$-model on the $2$-sphere associated with $(G,\mathbf M)$. In this first part, we check that the cohomology group has the correct graded dimensions expected from the monopole formula proposed by Cremonesi, Hanany and Zaffaroni arXiv:1309.2657. A ring structure (on the cohomology of a modified moduli space) will be introduced in the sequel of this paper.

math-ph↗
arXiv2016

Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras

We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quantized Coulomb branch is a deformation of a subquotient of the Yangian of the affine $\mathfrak{gl}(1)$.

math.RT↗
arXiv2014

Affine cellularity of quantum affine algebras

Cui (arXiv:1405.6441) has shown that the modified quantum affine algebra $\widetilde{\mathbf U}$ (more precisely its quotients, BLN algebras) is affine cellular in the sense of Koenig and Xi. The proof is based on the structure of cells of $\widetilde{\mathbf U}$, studied previously in [http://arxiv.org/abs/math/0212253], the author's joint work with Beck. We here give more direct proof based on Lemma 6.17 in [http://arxiv.org/abs/math/0212253], together with a property of the bilinear form introduced in [http://arxiv.org/abs/math/0204183]. We also prove that cell ideals are idempotent, and hence Theorem 4.4 in [Koenig-Xi] is applicable.

math.QA↗
arXiv2014

More lectures on Hilbert schemes of points on surfaces

This paper is based on author's lectures at Kyoto University in 2010 Summer, and in the 6th MSJ-SI `Development of Moduli Theory' at RIMS in June 2013. The purpose of lectures was to review several results on Hilbert schemes of points which were obtained after author's lecture note was written. Among many results, we choose those which are about equivariant homology groups $H^T_*(X^{[n]})$ of Hilbert schemes of points on the affine plane $X = \mathbb C^2$ with respect to the torus action.

math.RT↗
arXiv2013

Refined Chern-Simons theory and Hilbert schemes of points on the plane

Aganagic and Shakirov propose a refinement of the SU(N) Chern-Simons theory for links in three manifolds with S^1-symmetry, such as torus knots in S^3, based on deformation of the S and T matrices, originally found by Kirillov and Cherednik. We relate the large N limit of the S matrix to the Hilbert schemes of points on the affine plane. As an application, we find an explicit formula for the Euler characteristics of the universal sheaf, applied arbitrary Schur functor.

math.AG↗
arXiv2013

Cluster algebras and singular supports of perverse sheaves

We propose an approach to Geiss-Leclerc-Schroer's conjecture on the cluster algebra structure on the coordinate ring of a unipotent subgroup and the dual canonical base. It is based on singular supports of perverse sheaves on the space of representations of a quiver, which give the canonical base.

math.QA↗
arXiv2012

Quiver varieties and tensor products, II

We define a family of homomorphisms on a collection of convolution algebras associated with quiver varieties, which gives a kind of coproduct on the Yangian associated with a symmetric Kac-Moody Lie algebra. We study its property using perverse sheaves

math.QA↗
arXiv2011

Handsaw quiver varieties and finite W-algebras

Following Braverman-Finkelberg-Feigin-Rybnikov (arXiv:1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W-algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang-Mills theory with surface operators. Our new observation is that the C^*-fixed point set of a handsaw quiver variety is isomorphic to a graded quiver variety of type A, which was introduced by the author in connection with the representation theory of a quantum affine algebra. As an application, simple modules of the W-algebra are described in terms of IC sheaves of graded quiver varieties of type A, which were known to be related to Kazhdan-Lusztig polynomials. This gives a new proof of a conjecture by Brundan-Kleshchev on composition multiplicities on Verma modules, which was proved by Losev, in a wider context, by a different method.

math.QA↗
arXiv2010

Counting invariant of perverse coherent sheaves and its wall-crossing

We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi-Yau 3-folds and consider their Donaldson-Thomas type counting invariants. The stability depends on the choice of a component (= a chamber) in the complement of finitely many lines (= walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of chambers when the Calabi-Yau is the resolved conifold. For suitable choices of chambers, our invariants are specialized to Donaldson-Thomas, Pandharipande-Thomas and Szendroi invariants.

math.AG↗
arXiv2010

Quiver varieties and cluster algebras

Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig's dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [arXiv:math/0104151] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of `prime' simple ones according to the cluster expansion.

math.QA↗
arXiv2010

Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing

In earlier papers arXiv:0802.3120, arXiv:0806.0463 of this series we constructed a sequence of intermediate moduli spaces $\bM^m(c)$ connecting a moduli space $M(c)$ of stable torsion free sheaves on a nonsingular complex projective surface and $\bM(c)$ on its one point blow-up. They are moduli spaces of perverse coherent sheaves on the blow-up. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from $\bM^m(c)$ to $\bM^{m+1}(c)$, and then from $M(c)$ to $\bM(c)$. As an application we prove that Nekrasov-type partition functions satisfy certain equations which determine invariants recursively in second Chern classes. They are generalization of the blow-up equation for the original Nekrasov deformed partition function for the pure N=2 SUSY gauge theory, found and used to derive the Seiberg-Witten curves in arXiv:math/0306198.

math.AG↗
arXiv2010

Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting

We propose an explicit formula connecting Donaldson invariants and Seiberg-Witten invariants of a 4-manifold of simple type via Nekrasov's deformed partition function for the N=2 SUSY gauge theory with a single fundamental matter. This formula is derived from Mochizuki's formula, which makes sense and was proved when the 4-manifold is complex projective. Assuming our formula is true for a 4-manifold of simple type, we prove Witten's conjecture and sum rules for Seiberg-Witten invariants (superconformal simple type condition), conjectured by Mariño, Moore and Peradze.

math.DG↗