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 ℝ⁴ 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 ℝ⁴, 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. Mathematics Subject Classification (2000) 14D21, 57R57, 81T13, 81T60     

Asymptotic behaviour and the moduli space of doubly-periodic instantons

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line ℂ, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T×ℙ¹. The converse statement is also true, namely a holomorphic bundle on T×ℙ¹ which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperk?hler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperk?hler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus. Received June 8, 2000 / final version received February 1, 2001?Published online April 3, 2001     

Moduli of Quaternionic Superminimal Immersions of 2-Spheres into Quaternionic Projective Spaces

The aim of this paper is to describe the moduli spaces of degree d quaternionic superminimal maps from 2-spheres to quaternionic projective spaces HPn. We show that such moduli spaces have the structure of projectivized fibre products and are connected quasi-projective varieties of dimension 2nd + 2n + 2. This generalizes known results for spaces of harmonic 2-spheres in S⁴.     

Some results on the moduli spaces of quiver bundles

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.      

Exact construction of noncommutative instantons

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and give some exact instanton solutions for various noncommutative settings. We also present a new formula which is crucial to show an origin of the instanton number for U(1) and to prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data.     

Moduli spaces of meromorphic connections and quiver varieties

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 problem.     

Calabi–Yau Threefolds and Moduli of Abelian Surfaces I

We describe birational models and decide the rationality/unirationality of moduli spaces A _d (and A _d ^lev ) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.     

On quiver varieties and affine Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi?, M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ<Superscript>3</Superscript>

Symplectic instanton vector bundles on the projective space ℙ³ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I _n;r of rank-2r symplectic instanton vector bundles on ℙ³ with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I _n;r ^* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).     

On the Euler characteristic of Kronecker moduli spaces

Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.     

Intrinsic characterizations of Besov spaces on Lipschitz domains

The aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ? ?ⁿ is a bounded Lipschitz open subset in ?ⁿ. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ?ⁿ. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ?ⁿ described in Theorem 2.4 to the case of Lipschitz domains.     

Instantons and the Information Metric

The information metric arises in statistics as a natural inner product on a space of probability distributions. In general this inner product is positive semi-definite but is potentially degenerate.By associating to an instanton its energy density, we can examine the information metric g on the moduli spaces M of self-dual connections over Riemannian four-manifolds. Compared with the more widely known L² metric, the information metric better reflects the conformal invariance of the self-dual Yang–Mills equations, and seems to have better completeness properties. In the case of SU(2) instantons on S⁴ of charge one, g is known to be the hyperbolic metric on the five-ball. We show more generally that for charge-one SU(2) instantons over 1-connected, positive-definite manifolds, g is non-degenerate and complete in the collar region of M, and is asymptotically hyperbolic there; g vanishes at the cone points of M. We give explicit formulae for the metric on the space of instantons of charge one on CP².     

Local Quivers and Stable Representations

Abstract

In this paper we introduce and study the local quiver as a tool to investigate the étale local structure of moduli spaces of θ-semistable representations of quivers. As an application we determine the dimension vectors associated to irreducible representations of the torus knot groups G _p,q  = ?a, b ∣ a  ^p  = b ^q ?.     

Spin structures on instanton moduli spaces

We determine the necessary and sufficient conditions for the based SU(l)-instanton moduli spaces over CP² and the unbased SU(2)-instanton moduli spaces over S⁴ having spin structures.     

Mathematical Instantons in Characteristic Two

On P³, we show that mathematical instantons in characteristic two are unobstructed. We produce upper bounds for the dimension of the moduli space of stable rank two bundles on P³ in characteristic two. In cases where there is a phenomenon of good reduction modulo two, these give similar results in characteristic zero. We also give an example of a nonreduced component of the moduli space in characteristic two.     

Kac’s conjecture from Nakajima quiver varieties

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.     

The supersingular locus in Siegel modular varieties with Iwahori level structure

We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl–Oort stratification on the former, the Kottwitz–Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case g is even.     

Limit Theorems for Non-Commutative Random Variables

We study the weak law of large numbers and the central limit theorem for non-commutative random variables. We first define the concepts of variance and expectation for probability measures on homogeneous spaces, and formulate the weak law of large numbers and the central limit theorem for probability measures on locally compact groups. Then, we consider the non-commutative case, where the homogeneous space is replaced by a C^*-algebra that is equipped with a locally compact group G of automorphisms. We define the concepts of variance and expectation in the non-commutative situation. Furthermore, we prove that the weak law of large numbers and the central limit theorem hold for non-commutative random variables on if they hold on the group G of automorphisms.