Quaternionic Structures in Mathematics and Physics

The following sections are included:

• FOREWORD

• INTRODUCTION TO THE CONTRIBUTIONS

• List of Participants

• CONTENTS

The following sections are included:

• INTRODUCTION

• A SPECIAL HYPER-HERMITIAN METRIC

• HYPERCOMPLEX STRUCTURES ON CERTAIN NILPOTENT AND SOLVABLE LlE GROUPS

• REFERENCES

We generalize the hyperkähler quotient construction to the situation where there is no group action preserving the hyperkähler structure but for each complex structure there is an action of a complex group preserving the corresponding complex symplectic structure. Many (known and new) hyperkähler manifolds arise as quotients in this setting. For example, all hyperkähler structures on semisimple coadjoint orbits of a complex semisimple Lie group G arise as such quotients of T*G. The generalized Legendre transform construction of Lindström and Roček is also explained in this framework.

This article is a survey on the notion of quaternionic contact structures, which I defined in [2]. Roughly speaking, quaternionic contact structures are quaternionic analogues of integrable CR structures.

Let V = ℝ^p,q be the pseudo-Euclidean vector space of signature (p, q), p ≥ 3 and W a module over the even Clifford algebra Cℓ⁰(V). A homogeneous quaternionic manifold (M, Q) is constructed for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map II: ∧²W → V. If the skew symmetric vector valued bilinear form II is nondegenerate then (M, Q) is endowed with a canonical pseudo-Riemannian metric g such that (M, Q, g) is a homogeneous quaternionic pseudo-Kähler manifold. If the metric g is positive definite, i.e. a Riemannian metric, then the quaternionic Kähler manifold (M, Q, g) is shown to admit a simply transitive solvable group of automorphisms. In this special case (p = 3) we recover all the known homogeneous quaternionic Kähler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If p > 3 then M does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kähler manifolds. Then it is shown that for q = 0 the noncompact quaternionic manifold (M, Q) can be endowed with a Riemannian metric h such that (M, Q, h) is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if p > 3.

The twistor bundle Z → M and the canonical SO(3)-principal bundle S → M associated to the quaternionic manifold (M, Q) are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution of complex codimension one, which is a complex contact structure if and only if II is nondegenerate. Moreover, an equivariant open holomorphic immersion into a homogeneous complex manifold of complex algebraic group is constructed.

Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map II : ∨²W → V a homogeneous quaternionic supermanifold (M, Q) is constructed and, moreover, a homogeneous quaternionic pseudo-Kähler supermanifold (M, Q, g) if the symmetric vector valued bilinear form II is nondegenerate.

Almost-complex and hyper-complex manifolds are considered in this paper from the point of view of complex analysis and potential theory. The idea of holomorphic coordinates on an almost-complex manifold (M, J) is suggested by D. Spencer [Sp]. For hypercomplex manifolds we introduce the notion of hyperholomorphic function and develop some analogous statements. Elliptic equations are developed in a different way than D. Spencer. In general here we describe only the formal aspect of the developed theory.

This note is a revised version of the talk given by the author at the meeting Quaternionic structures in Mathematics and Physics at Rome in September, 1999. The results presented here are part of [4], a joint work with R. Miatello.

Let ℍ be the algebra of quaternions generated by e₁, e₂ and e₁₂ satisfying e₁e₂ = e₁₂ and e_ie_j + e_je_i = -2δ_ij for i = 1, 2, 12. Any element x in ℍ ⊕ ℍ may be decomposed as x = Px + Qxe₃ for quaternions Px and Qx. The generalized Cauchy-Riemann operator in ℝ⁴ is defined by . Leutwiler noticed that the power function (x₀ + x₁e₁ + x₂e₂ + x₃e₃)^m is the solution of the generalized Cauchy-Riemann system x₃Df + 2f₃ = 0 which has connections to the hyperbolic metric. We study solutions of the equation x₃Df + 2Q' (f) = 0 (the prime ' is the main involution) called hyperholomorhic functions. If f = f₀ + f₁e₁ + f₂e₂ + f₃e₃ for some real functions f₀, f₁, f₂, f₃ then f is the solution of the generalized Cauchy-Riemann system stated earlier.

This is a report on work in process. We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to Kähler reduction by considering the Kähler cone over a Sasakian manifold. Fianlly, we present an example of Sasakian manifold obtained by SU(2) reduction of a standard Sasakian sphere.

The following sections are included:

• INTRODUCTION
  • Quaternionic tensor products
  • Stable and semistable Aℍ-modules
  • Stable Aℍ-modules and exact sequences

• ALGEBRAIC STRUCTURES OVER THE QUATERNIONS
  • H-algebras and modules
  • H-algebras and Poisson brackets
  • Filtered and graded H-algebras
  • Free and finitely-generated H-algebras

• HYPERCOMPLEX GEOMETRY
  • Q-holomorphic functions on hypercomplex manifolds
  • Hypercomplex manifolds and H-algebras
  • Hyperkähler manifolds and HP-algebras
  • Reconstructing a hypercomplex manifold from its H-algebra
  • Asymptotically conical hyperkähler manifolds

• EXAMPLES, APPLICATIONS AND CONCLUSIONS
  • Q-holomorphic functions on ℍ
  • Hypercomplex manifolds undetermined by their H-algebras
  • HL-algebras and HP-algebras
  • Hyperkähler manifolds with symmetries
  • The Eguchi-Hanson space
  • Self-dual vector bundles and H-algebra modules
  • Directions for future research

• REFERENCES

A canonical hyperkähler metric on the total space T*M of a cotangent bundle to a complex manifold M has been constructed recently by the author in [K]. This paper presents the results of [K] in a streamlined and simplified form. The only new result is an explicit formula obtained for the case when M is an Hermitian symmetric space.

A smooth hyper Kähler quotient of a quaternionic vector space H^N by a subtorus of T^N is called a toric hyperKähler manifold. We determine the ring structure of the integral equivariant cohomology of a toric hyperKähler manifold.

R. Penrose has observed in 1976 [11] that the points of the Minkowski space-time can be represented by two-dimensional linear subspaces of a complex four-dimensional vector space on which an hermitian form of signature (++,--) is defined. He called this flat twistor space, and the method of investigating deformation of complex structures, yielded from there, the twistor programme. This initiated a series of papers and monographs by various authors. In the present research we are dealing with dynamical systems generated by the Hermitian Hurwitz pairs of the signature (σ, s), σ + s = 5 + 4µ, |σ + 1 - s| = 2 + 4m; µ, m = 0, 1, … In particular, for (3,2) and its dual (1,4) the role of entropy is indicated as well as the relationship between Hurwitz and Penrose twistors; Hurwitz twistors being objects introduced by us. The signatures (1,8) and (7,6) give rise for introducing pseudotwistors and bitwistors, respectively; for pseudotwistors we can prove [9] a counterpart of the original fundamental Penrose theorem in the local version (on real analytic solutions of the related spinor equations vs. harmonic forms) and in the semi-global version (on holomorphic solutions of those equations vs. Dolbeault cohomology groups). This has to be preceded by basic constructions (which is the core topic of this paper), a study of the related pseudotwistors and spinor equations as well as complex structures on spinors. This will allow us to prove a theorem (which we call the atomization theorem) saying that there exist complex structures on isometric embeddings for the Hermitian Hurwitz pairs concerned so that the embeddings are real parts of holomorphic mappings.

It is well-known that all geodesics in a Riemannian symmetric space of rank one are congruent each other under the action of isometry group. In this paper we are interested in circles in a quaternionic projective space ℚPⁿ. Recently we have known that each circle in ℚPⁿ is congruent to a circle in ℂPⁿ which is a totally geodesic submanifold of ℚPⁿ. This fact leads us to the study about circles in ℂPⁿ.

In the paper special 4-planar mappings of almost Hermitian quaternionic spaces are studied. Fundamental equations of these mappings are expressed in linear Cauchy form. Our results improve results of I.N. Kurbatova [9].

The following sections are included:

• INTRODUCTION

• A SHORT REVIEW ON SPIN AND SPIN^c GEOMETRY

• RELATIONS BETWEEN COMPLEX CONTACT STRUCTURES AND SPINORS

• THE CLASSIFICATION OF POSITIVE KÄHLER-EINSTEIN CONTACT MANIFOLDS

• REFERENCES

The following sections are included:

• Introduction

• PRELIMINARIES
  • The Wolf spaces and the Salamon twistor spaces
  • ASD-connection

• HOMOGENEOUS ASD BUNDLES

• MONAD AND REPRESENTATION THEORY
  • The unified construction of monad

• MODULI SPACES
  • Description of moduli

• SINGULAR SETS

• REFERENCES

We study Sp(1)ⁿ-invariant hyperKähler or quaternionic Kähler manifolds of real dimension 4n. In the case of n = 1, Hitchin classified these kinds of metrics associated with special functions…

The investigation of strings and M-theory involves the understanding of various BPS solitons which in a certain approximation can be thought of as solutions of ten- and eleven-dimensional supergravity theories. These solitons have a brane or a intersecting brane interpretation, saturate a bound and are associated with parallel spinors with respect to a connection of the spin bundle of spacetime. A class of intersecting brane configurations is examined and it is shown that the geometry of spacetime is hyper-Kähler with torsion. A relation between these hyper-Kähler geometries with torsion and quaternionic calibrations is also demonstrated.

The following sections are included:

• INTRODUCTION

• KÄHLER-WEYL 4-MANIFOLDS

• LIE GROUPS AND HYPERCOMPLEX GEOMETRY

• THE SWANN BUNDLE

• REFERENCES

This article is a revised version of the author's lecture given at the Second Meeting on Quaternionic Structures in Mathematics and Physics at Roma, September 8, 1999. It will appear in the proceeding of this conference. The work presented here are jointly done with Gueo Grantcharov. This article is composed while the author visits the ESI.

In this paper we apply our recent geometric theory of noncommutative (quantum) manifolds and noncommutative (quantum) PDEs [7,8,12] to the category of quantum quaternionic manifolds. These are manifolds modelled on spaces built starting from quaternionic algebras. For PDEs considered in such category we determine theorems of existence of local and global quantum quaternionic solutions. We shaw also that such a category of quantum quaternionic manifolds properly contains that of manifolds with (almost) quaternionic structure. So our theorems of existence of quantum quaternionic manifolds for PDEs produce a cascade of new solutions with nontrivial topology.

An optimal control problem on the Lie group SP (1) is discussed and some of its dynamical and geometrical properties are pointed out.

A weight system on graph homology was constructed by Rozansky and Witten using a compact hyperkähler manifold. A variation of this construction utilizing holomorphic vector bundles over the manifold gives a weight system on chord diagrams. We investigate these weights from the hyperkähler geometry point of view.

We study quaternionic group representations of finite groups systematically and obtain some basic tools of the theory, such as orthogonality relations and the Clabsch-Gordan series for reducible representations. We also derive all irreducible inequivalent Q-representations of a group G, classifying them according to a suitable generalization of the Wigner and the Frobenius-Schur classification. Some applications to physical problems and to the time reversal symmetry are shown.

In this article we discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenböck formula for the Hodge–Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For quaternionic Kähler manifolds this leads to simple proofs of eigenvalue estimates for Dirac and Laplace operators. We determine which representations may contribute to harmonic forms and prove the vanishing of certain odd Betti numbers on compact quaternionic Kähler manifolds of negative scalar curvature. We simplify the proofs of several related results in the positive case.

These notes are based on a I talk I gave at the Erwin Schrödinger International Institute for Mathematical Physics, Vienna, on the 20th October, 1999. This is work in progress, partly based on joint work with F. M. Cabrera and M. D. Monar and partly results of my Ph. D. student Richard Cleyton. It is a pleasure to thank the Erwin Schrödinger Institute and the organisers of the program on Holonomy Groups in Differential Geometry for their kind hospitality.

We present two results that we have not found in the literature and that we believe therefore to be new, and some of their consequences. First, the Maxwell equations and the Lorentz force are formulated a with strict use of Hamilton's quaternions (two quaternion field equations and one quaternion force equation). Second, formulas for the Lorentz transformation, in fact for the 15 parameter conformal group, are presented, again with strict use of Hamilton's quaternions…

The geometry that is defined by the scalars in couplings of Einstein–Maxwell theories in N = 2 supergravity in 4 dimensions is denoted as special Kähler geometry. There are several equivalent definitions, the most elegant ones involve the symplectic duality group. The original construction used conformal symmetry, which immediately clarifies the symplectic structure and provides a way to make connections to quaternionic geometry and Sasakian manifolds.

This is a survey of some of the work done in 1993-99 on resolution of singularities in the context of hyperkähler geometry. We define a singular hypercomplex variety and its desingularization; similar methods are applied to desingularising coherent sheaves. We relate the singularities of reflexive sheaves over hyperkähler manifolds to quaternionic-Kähler geometry. Finally, we study holomorphic symplectic orbifolds and their resolutions.