Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

@article{Nakajima1994InstantonsOA,
  title={Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras},
  author={Hiraku Nakajima},
  journal={Duke Mathematical Journal},
  year={1994},
  volume={76},
  pages={365-416}
}
  • H. Nakajima
  • Published 1 November 1994
  • Mathematics
  • Duke Mathematical Journal
To Professor Shoshichi Kobayashi on his 60th birthday 1. Introduction. In this paper we shall introduce a new family of varieties, which we call quiver varieties, and study their geometric structures. They have close relation to the singularity theory and the representation theory of the Kac-Moody algebras. Our original motivation was to study solutions of the anti-self-dual Yang-Mills equations on a particular class of 4-dimensional noncompact complete manifolds, the so-called ALE spaces (or… 

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References

SHOWING 1-10 OF 79 REFERENCES

Homology of moduli spaces of instantons on ALE spaces. I

In [13] P. B. Kronheimer and the author introduced a new class of hyper-Kahler manifolds which arise as moduli spaces of anti-self-dual connections on a certain class of 4-dimensional noncompact

Instantons and affine algebras I: The Hilbert scheme and vertex operators

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex

On crystal bases of the $Q$-analogue of universal enveloping algebras

0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the

Quivers, perverse sheaves, and quantized enveloping algebras

1. Preliminaries 2. A class of perverse sheaves on Ev 3. Multiplication 4. Restriction 5. Fourier-Deligne transform 6. Analysis of a sink 7. Multiplicative generators 8. Compatibility of

Langlands Reciprocity for Algebraic Surfaces

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface.

Closures of conjugacy classes of matrices are normal

Hanspeter Kraft 1 and Claudio Procesi 2 Sonderforschungsbereich Theoretische Mathematik, Universit~it Bonn D-5300 Bonn, Federal Republic of Germany 2 Istituto di Matematica, Universitfi di Roma,

THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE

In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean

Stratified symplectic spaces and reduction

Let (M, w) be a Hamiltonian G-space with proper momentum map J: M -> g*. It is well-known that if zero is a regular value of J and G acts freely on the level set J '(0), then the reduced space MO =

Instability in invariant theory

Let V be a representation of a reductive group G. A fundamental theorem in geometric invariant theory states that there are enough polynomial functions on V, which are invariant under G, to
...