The meeting "Quaternionic Structures in Mathematics and
Physics" took place at the International School for Advanced Studies
(SISSA), Trieste, 5-9 September 1994, under the sponsorship of its
Interdisciplinary Laboratory. The purpose of the meeting was to bring
together scientists from different areas of mathematics and physics
working in the field of quaternionic structures. New results were
presented in the theory of quaternionic manifolds, together with progress
in Clifford analysis and supersymmetry.
Interest for the scientific encounter was stronger than
had been expected, and by popular demand the organizers decided to invite
the speakers to contribute to what has become the present volume. The
aim was to present recent advances to those working in the relevant
sectors of geometry, analysis and theoretical physics.
On behalf of the Organizing Committee, we wish to thank SISSA and the
Interdisciplinary Laboratory for their kind hospitality and financial
support, the European Mathematical Society for having helped to cover the
expenses of two speakers from Eastern Europe, and the Mathematics
Commitee of the Consiglio Nazionale delle Ricerche for support in the
preparation of a preliminary version of this volume which appeared as a
SISSA preprint, ILAS/FM-6/1996. We also owe a debt of gratitude to Rosanna
Sain and Rosanna de Iurco for their invaluable help, assistance and
patience of the present volume and to Angelo Bardelloni for the technical
preparation of the electronic version of these Proceedings.
Finally, it gives us great pleasure to thank all the partecipants of the
meeting for their interest and enthusiasm, and not least the
contributors and the referees of the present volume.
ORGANIZING COMMITTEE OF THE MEETING:
D. Alekseevsky, E.
Bonan, S. Cecotti, P. Fré, G. Gentili, S. Marchiafava, M.
Pontecorvo, S. Salamon, G. Tomassini.
INTRODUCTION TO THE CONTRIBUTIONS
Given the importance of complex analysis and the concept of a
complex manifold, it is not surprising that a great deal of effort has
been made to develop analogous theories based upon the quaternions.
Advances in the theory of four-dimensional geometry is one of the factors
that has led to the successful exposition and development of various
types of quaternionic manifolds, all of which attempt to generalize in
some way structures in dimension 4 to dimensions 4n where n>1.
Independent progress in analysis related to the Dirac operator has led to
a greater understanding of the definition and properties of functions
possessing some form of quaternionic derivative, and one is beginning to
see the analytical and geometrical aspects of the subject unite to serve
From the point of view of homogeneous geometry, quaternion-Kähler
manifolds form a natural class to investigate. Alekseevsky did a lot to
get the subject underway by describing a class of Einstein homogeneous
spaces with negative scalar curvature, corresponding to certain solvable
Lie algebras. The more recent discovery that these spaces play a
fundamental role in the classification of supergravity theories is
explained in the paper by de Wit and Van Proeyen, who discuss them in
parallel with the class of special Kähler manifolds that arise in
connection with Calabi-Yau 3-folds. A detailed and complete classification
of Alekseevsky spaces is carried out by Cortés from a different
point of view, and he shows that the symmetric examples are precisely
those with non-positive sectional curvature.
Nagano studies orbits of isotropy groups in Riemannian symmetric spaces,
which automatically highlights the class of compact quaternion-Kähler
manifolds. His approach unifies different types of geometrical structures
and constructs their twistor spaces that parametrize families of almost
complex structures. An algebro-geometric description of twistor spaces of
hyper-Kähler manifolds is given by Fujiki, interpreting and extending
the first known example of a non-projective compact algebraic manifold.
The theory of hyper-Kähler metrics, whilst arguably under-represented in
these proceedings, also underlies the paper of Swann, who investigates
nilpotent coadjoint orbits, quotients and invariant theory. The resulting
quaternionic structures are homogeneous for the action of a complex group,
and incorporate the compact complex contact spaces characterized by
Lichnerowicz in the context of Poisson manifolds and moment mappings. The
theory of Poisson manifolds is also the setting for the paper of
Fernández, Ibáñez and de Léon, who discuss
symplectically harmonic forms on a nilmanifold.
Quaternionic manifolds, through their associated bundles discussed in
papers already mentioned, give rise to a host of other geometries, some of
which are of equal, if not greater, importance. Such is the case for the
class of 3-Sasakian manifolds, described by Boyer, Galicki and Mann, whose
theory to some extent mirrors that of quaternion-Kähler orbifolds.
Whilst 3-Sasakian manifolds are Einstein, their holonomy does not reduce
and they are intriguing for this very reason. They recur in a
conformally-invariant setting in the paper by Ornea and Piccinni, who
characterize certain Weyl structures and evaluate Betti numbers of
associated hypercomplex manifolds.
The definition of anti-self-dual connections in higher dimensions pursues
the analogy with 4-dimensional geometry and theoretical physics, and their
moduli constitute a more radical type of auxiliary space that remains to
be fully investigated. This topic is introduced by Nagatomo, who proves a
number of vanishing theorems and then gives a monad description on the
complex Grassmannian. The algebra of almost complex structures is
exploited by Hangan to characterize Killing vector fields preserving a
quaternionic structure, and exterior algebra associated to the fundamental
4-form is the subject of the paper by Bonan. Differential forms
andgeneralizations of self-duality are also used in the paper by
Fernández and Ugarte on the de Rham cohomology of 7-manifolds with
a structure determined by the exceptional group G2.
Clifford algebras and the Dirac operator form the bridge between the
geometry and analysis represented by the contributions to the meeting's
proceedings. Clifford algebras crop up in supergravity and other topics
mentioned above, and the determination of the eigenvalues of the Dirac
operator over compact manifolds with reduced holonomy is an active
problem, addressed by Hijazi. The related concept of Killing spinor also
occurs in the characterization of twistor spaces given by Moroianu and
Semmelmann. On the other hand, various types of monogenic functions are
defined as solutions of a suitable Dirac operator, and in dimension 4
correspond to the quaternionic holomorphic functions introduced by Fueter
and described in Sudbery's well-known paper.
Conformal invariance of the Dirac operator is the starting point for Ryan,
whose paper introduces Clifford analysis and the theory of monogenic
functions. He proceeds to define a Fourier transform for L² functions
over spheres with values in a complex Clifford algebra, and considers
applications involving the resulting convolution and projection operators.
Integral formulas for the Dirac operator form the basis of the paper by
Cnops, who defines the Dirac operator in terms of Stokes' formula, and
also considers spherical representations. Such analysis is an appropriate
setting for quaternionic function theory, though the latter has a special
flavour. Pernas adopts a direct approach to quaternionic holomorphic
functions, defining them in the setting of hypercomplex manifolds, and
then applying them to a study of hyperbolic space. An alternative
definition of quaternionic derivatives is presented by Lugojan with useful
references and examples.