Showing papers in "Duke Mathematical Journal in 1994"

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

Abstract: 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 the ALE gravitational instantons), which were constructed by Kronheimer [Krl]. In [KN] we gave a description of the framed moduli space of all solutions in terms of solutions of a system of quadratic equations (called the ADHM equations) for representations of a quiver on an affine, simply laced Dynkin graph. It is an analogue of the description, given by Atiyah, Drinfeld, Hitchin, and Manin [ADHM], of the moduli space for IR 4 (or S4) in terms of solutions of a quadratic equation for certain finite-dimensional matrices. Once we set aside their gauge-theoretic origin, there is no longer reason to restrict ourselves to affine Dynkin graphs. Definitions can be generalized to arbitrary finite graphs. We get what we call quiver varieties. We study geometric structures of quiver varieties in this paper. In [Nal] it was noticed that the moduli space of anti-self-dual connections on ALE spaces has a hyper-K/ihler structure, namely a Riemannian metric equipped with three endo-morphisms I, J, K of the tangent bundle which satisfy the relations of quaternion algebra and are covariant constant with respect to the Levi-Civita connection: The same holds for general quiver varieties. In particular, quiver varieties have holomorphic symplectic forms. We study further properties of the quiver variety, such as a natural *-action, symplectic geometry, topology, and so on. As ALE spaces closely related to simple singularities, quiver varieties have very special kinds of singularities that enjoy very nice properties. Surprisingly, the ADHM equation appears in a very different context. In [L3] Lusztig used it to construct \"canonical bases\" of the part U-of the quantized enveloping algebra U associated by Drinfeld and Jimbo to the graph. Motivated by his results, we give a geometric construction of irreducible highest-weight integrable representations of the Kac-Moody algebra associated to the graph (Theorem 10.14). The weight space of the representation space will be given as a vector space consisting of constructible functions on a Lagrangian subvariety of a quiver variety. The action of the Kac-Moody …

Koszul duality for operads

Abstract: (0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] and Kontsevich [Kon 1] on the cell decomposition and intersection theory on the moduli spaces of curves. The other is the theory of Koszul duality for quadratic associative algebras which was introduced by Priddy [Pr] and has found many applications in homological algebra, algebraic geometry and representation theory (see e.g., [Be] [BGG] [BGS] [Ka 1] [Man]). The unifying concept here is that of an operad. This paper can be divided into two parts consisting of chapters 1, 3 and 2, 4, respectively. The purpose of the first part is to establish a relationship between operads, moduli spaces of stable curves and graph complexes. To each operad we associate a collection of sheaves on moduli spaces. We introduce, in a natural way, the cobar complex of an operad and show that it is nothing but a (special case of the) graph complex, and that both constructions can be interpreted as the Verdier duality functor on sheaves. In the second part we introduce a class of operads, called quadratic, and introduce a distinguished subclass of Koszul operads. The main reason for introducing Koszul operads (and in fact for writing this paper) is that most of the operads ”arising from nature” are Koszul, cf. (0.8) below. We define a natural duality on quadratic operads (which is

Lie bialgebroids and Poisson groupoids

Abstract: Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a certain sense. For a Poisson group G, the multiplicative Poisson structure π induces a Lie algebra structure on the Lie algebra dual g∗ which makes (g, g∗) into a Lie bialgebra. In fact, there is a one-one correspondence between Poisson Lie groups and Lie bialgebras if the Lie groups are assumed to be simply connected [7], [16], [19]. The importance of Poisson Lie groups themselves arises in part from their role as classical limits of quantum groups [8] and in part because they provide a class of Poisson structures for which the realization problem is tractable [15]. Poisson groupoids were introduced by Weinstein [24] as a generalization of both Poisson Lie groups and the symplectic groupoids which arise in the integration of arbitrary Poisson manifolds [4], [11]. He noted that the Lie algebroid dual A∗G of a Poisson groupoid G itself has a Lie algebroid structure, but did not develop the infinitesimal structure further. In this paper we introduce and study a natural infinitesimal invariant for Poisson groupoids, the Lie bialgebroids of the title. Our ultimate purpose is to develop a Lie theory for Poisson groupoids parallel to that for Poisson groups. In this paper we are primarily concerned with the first half of this process; that is, with the construction of the Lie bialgebroid of a Poisson groupoid. After the preliminary §2, in which we describe the generalization to arbitrary Lie algebroids of the exterior calculus and Schouten calculus, in §3 we define a Lie bialgebroid to be a Lie algebroid A whose dual A∗ is also equipped with a Lie algebroid structure, such that the coboundary operator d∗ : A −→ ∧(A) associated to A∗ satisfies a cocycle equation with respect to Γ(A), the Lie algebra of sections of A. This is clearly a straightforward extension of the concept of a Lie bialgebra [16] but cannot be formalized in Lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general Lie algebroid. Most of §3 is devoted to proving that this definition is self-dual: if (A,A∗) is a Lie bialgebroid, then (A∗, A) is also. In §4, we briefly consider the special case of Lie bialgebroids satisfying a triangularity condition, which include some important examples such as the usual triangular Lie bialgebras and Lie bialgebroids associated to Poisson manifolds. The techniques used in §§2–4 are similar to those known for Lie bialgebras. It would be possible, by suitably generalizing the proof for Poisson groups, to prove

On the Hodge structure of projective hypersurfaces in toric varieties

Abstract: This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive cohomology of $X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$ and $T$ is an affine hypersurface in $T$, and we show how recent results of the second author can be stated using various ideals in the ring $S$. To prove our results, we must give a careful description (in terms of $S$) of $d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties. Other topics considered in the paper include quasi-smooth hypersurfaces and $V$-submanifolds, the structure of the complement of $U$ when $P$ is represented as the quotient of an open subset $U$ of affine space, a generalization of the Euler exact sequence on projective space, and the relation between graded pieces of $R/J(f)$ and the moduli of ample hypersurfaces in $P$.

A finiteness theorem for elliptic Calabi-Yau threefolds

Abstract: We prove that up to birational equivalence, there exists only a finite number of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical class and factorial terminal singularities) which have an elliptic fibration to a rational surface. This strengthens a result of B. Hunt that there are only a finite number of possible Euler characteristics for such threefolds.

Representations of affine Lie algebras, parabolic differential equations, and Lamé functions

Abstract: We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form $F=\Tr (\Phi_1 (z_1)\ldots \Phi_n (z_n)q^{-\d}e^{h})$ where $\Phi_i$ are intertwiners between Verma modules and evaluation modules over an affine Lie algebra $\ghat$, $\d$ is the grading operator in a Verma module and $h$ is in the Cartan subalgebra of $\g$. We derive a system of differential equations satisfied by such a function. In particular, the calculation of $q\frac{\d} {\d q} F$ yields a parabolic second order PDE closely related to the heat equation on the compact Lie group corresponding to $\g$. We consider in detail the case $n=1$, $\g = \sltwo$. In this case we get the following differential equation ($q=e^{\pi \i \tau}$): $ \left( -2\pi\i (K+2)\frac{\d}{\d\tau} +\frac{\d^2}{\d x^2}\right) F = (m(m+1)\wp(x+\frac{\tau}{2}) +c)F$, which for $K=-2$ (critical level) becomes Lam\'e equation. For the case $m\in\Z$ we derive integral formulas for $F$ and find their asymptotics as $K\to -2$, thus recovering classical Lam\'e functions.

Rank one elliptic $A$-modules and $A$-harmonic series

Abstract: 209. A9] G. Anderson: A double complex for computing the sign-cohomology of the universal ordinary distribution, (preprint). An1] B. Angl es: Sur les anneaux d'endomorphismes des modules de Drinfeld d eenis sur des corps nis, C.R.

Elliptic Dunkl operators, root systems, and functional equations

Abstract: We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases. In particular, solutions associated with elliptic curves are constructed. In the $A_{n-1}$ case, we discuss the relation with elliptic Calogero-Moser integrable $n$-body problems, and discuss the quantization ($q$-analogue) of our construction.