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Hiraku Nakajima

Publications and source records attributed to Hiraku Nakajima.

At least 55 records · Page 3Linked to original sources
arXiv2009

Quiver Varieties and Branching

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group $G_\aff$ [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of $G_{\mathrm{cpt}}$-instantons on $\R^4/\Z_r$ correspond to weight spaces of representations of the Langlands dual group $G_\aff^\vee$ at level $r$. When $G = \SL(l)$, the Uhlenbeck compactification is the quiver variety of type $\algsl(r)_\aff$, and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for $G=\SL(l)$.

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arXiv2008

Perverse coherent sheaves on blow-up. I. a quiver description

This is the first of two papers studying moduli spaces of a certain class of coherent sheaves, which we call {\it stable perverse coherent sheaves}, on the blowup of a projective surface. They are used to relate usual moduli spaces of stable sheaves on a surface and its blowup. In this first part, we give a quiver (or ADHM) description of moduli spaces for framed perverse coherent sheaves on the blowup of the projective plane.

math.AG↗
arXiv2008

Perverse coherent sheaves on blow-up. II. wall-crossing and Betti numbers formula

This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up of a projective surface at a point. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle associated with the exceptional divisor. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in arXiv:0802.3120: c) The moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space of stable coherent sheaves on the original surface if the first Chern class is orthogonal to the exceptional divisor. d) The moduli space becomes isomorphic to the usual moduli space of stable coherent sheaves on the blow-up after twisting by sufficiently large negative power of the line bundle associated with the exceptional curve. Therefore usual moduli spaces of stable sheaves on the blow-up and the original surfaces are connected via wall-crossings.

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arXiv2006

K-theoretic Donaldson invariants via instanton counting

In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms.

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arXiv2006

Instanton counting and Donaldson invariants

For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture math.AG/0306198, hep-th/0306238, math.AG/0409441 and its refinement math.AG/0311058, we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with $b_+=1$ in terms of modular forms. This formula was proved earlier in alg-geom/9506018 more generally for simply connected 4-manifolds with $b_+=1$, assuming the Kotschick-Morgan conjecture and it was also derived by physical arguments in hep-th/9709193.

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arXiv2006

$t$--analogs of $q$--characters of quantum affine algebras of type $E_6$, $E_7$, $E_8$

We compute $t$--analogs of $q$--characters of all $l$--fundamental representations of the quantum affine algebras of type $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ by a supercomputer. In particular, we prove the fermionic formula for Kirillov-Reshetikhin modules conjectured by Hatayama et al. (math.QA/9812022) for these classes of representations. We also give explicitly the monomial realization of the crystal of the corresponding fundamental representations of the qunatum enveloping algebras associated with finite dimensional Lie algebras of types $E_6$, $E_7$, $E_8$. These are computations of Betti numbers of graded quiver varieties, quiver varieties and determination of all irreducible components of the lagrangian subvarities of quiver varieties of types $E_6$, $E_7$, $E_8$ respectively.

math.QA↗
arXiv2006

Level 0 Monomial crystals

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of $E_6^{(2)}$, $E_7^{(1)}$, $E_8^{(1)}$. Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

math.QA↗
arXiv2005

Instanton counting on blowup. II. $K$-theoretic partition function

We study Nekrasov's deformed partition function of 5-dimensional supersymmetric Yang-Mills theory compactified on a circle. Mathematically it is the generating function of the characters of the coordinate rings of the moduli spaces of instantons on $\mathbb R^4$. We show that it satisfies a system of functional equations, called blowup equations, whose solution is unique. As applications, we prove (a) logarithm of the partition function times $ε_1ε_2$ is regular at $ε_1 = ε_2 = 0$, (a part of Nekrasov's conjecture), and (b) the genus 1 parts, which are first several Taylor coefficients of the logarithm of the partition function, are written explicitly in terms of the Seiberg-Witten curves in rank 2 case.

math.AG↗
arXiv2005

Instanton counting on blowup. I. 4-dimensional pure gauge theory

We give a mathematically rigorous proof of Nekrasov's conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$ gives a deformation of the Seiberg-Witten prepotential for N=2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of $\mathbb R^4$, we derive a differential equation for the Nekrasov's partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al.

math.AG↗
arXiv2003

Lectures on Instanton Counting

These notes have two parts. The first is a study of Nekrasov's deformed partition functions $Z(\ve_1,\ve_2,\vec{a};\q,\vecτ)$ of N=2 SUSY Yang-Mills theories, which are generating functions of the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$. The second is review of geometry of the Seiberg-Witten curves and the geometric engineering of the gauge theory, which are physical backgrounds of Nekrasov's partition functions. The first part is continuation of math.AG/0306198, where we identified the Seiberg-Witten prepotential with $Z(0,0,\vec{a};\q,0)$. We put higher Casimir operators to the partition function and clarify their relation to the Seiberg-Witten $u$-plane. We also determine the coefficients of $\ve_1\ve_2$ and $(\ve_1^2+\ve_2^2)/3$ (the genus 1 part) of the partition function, which coincide with two measure factors $A$, $B$ appeared in the $u$-plane integral. The proof is based on the blowup equation which we derived in the previous paper.

math.AG↗
arXiv2002

Crystal bases and two-sided cells of quantum affine algebras

Let $\g$ be an affine Kac-Moody Lie algebra. Let $\U^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\g$. We construct a basis of $\U^+$ which is related to the Kashiwara-Lusztig global crystal basis (or canonical basis) by an upper triangular matrix (with respect to an explicitly defined ordering) with 1's on the diagonal and with above diagonal entries in $q_s^{-1} \Z[q_s^{-1}]$. Using this construction we study the global crystal basis $\B(\Um)$ of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl like decomposition of the crystal $\B(\Um)$ (Theorem 4.18), as well as an explicit description of two-sided cells of $\B(\Um)$ and the limit algebra of $\Um$ at $q=0$ (Theorem 6.45).

math.QA↗
arXiv2002

Geometric construction of representations of affine algebras

Let $Γ$ be a finite subgroup of $\SL_2(\C)$. We consider $Γ$-fixed point sets in Hilbert schemes of points on the affine plane $\C^2$. The direct sum of homology groups of components has a structure of a representation of the affine Lie algebra $\ag$ corresponding to $Γ$. If we replace homology groups by equivariant $K$-homology groups, we get a representation of the quantum toroidal algebra $\Ut$. We also discuss a higher rank generalization and character formulas in terms of intersection homology groups.

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arXiv2002

Extremal weight modules of quantum affine algebras

Let $\ag$ be an affine Lie algebra, and let $\Ua$ be the quantum affine algebra introduced by Drinfeld and Jimbo. In [Kas94] Kashiwara introduced a $\Ua$-module $V(λ)$, having a global crystal base for an integrable weight $λ$ of level 0. We call it an {\it extremal weight module}. In [\S13]{Kas00} Kashiwara gave a conjecture on the structure of extremal weight modules. We prove his conjecture when $\ag$ is an untwisted affine Lie algebra of a simple Lie algebra $\g$ of type $ADE$, using a result of Beck-Chari-Pressley [BCP].

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arXiv2002

Quiver varieties and t-analogs of q-characters of quantum affine algebras

Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ``computable'' polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ``compute'' $q$-characters for all simple modules. The result is based on ``computations'' of Betti numbers of graded/cyclic quiver varieties. (The reason why we put `` '' will be explained in the end of the introduction.)

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arXiv2001

Quiver varieties and tensor products

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety $\Zl$ in a quiver variety, and show the following results: (1) The homology group of $\Zl$ is a representation of a symmetric Kac-Moody Lie algebra $\mathfrak g$, isomorphic to the tensor product $V(λ_1)\otimes...\otimes V(λ_N)$ of integrable highest weight modules. (2) The set of irreducible components of $\Zl$ has a structure of a crystal, isomorphic to that of the $q$-analogue of $V(λ_1)\otimes...\otimes V(λ_N)$. (3) The equivariant $K$-homology group of $\Zl$ is isomorphic to the tensor product of universal standard modules of the quantum loop algebra $\Ul$, when $\mathfrak g$ is of type $ADE$. We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.

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