Quasi-Conformal Actions,Quaternionic Discrete Series and Twistors: <Emphasis Type="Italic">SU</Emphasis>(2, 1) and <Emphasis Type="Italic">G</Emphasis><Subscript>2(2)</Subscript>

Quasi-conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane, to Hermitian symmetric tube domains based on arbitrary Jordan algebras, and further to arbitrary Freudenthal triple systems. In the mathematics literature, quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionic-Kähler spaces. These two constructions are essentially identical, as we show explicitly for the two rank 2 cases SU(2, 1) and G ₂₍₂₎. We obtain explicit results for certain principal series, quaternionic discrete series and minimal representations of these groups, including formulas for the lowest K-types in various polarizations. We expect our results to have applications to topological strings, black hole micro-state counting and to the theory of automorphic forms.     

Quaternionic Lorentz Group and Dirac Equation

We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation.     

Notes on quaternionic group representations

We study quaternionic group representations of finite groups systematically and obtain some basic tools of the theory, such as orthogonality relations and the Clebsch-Gordan series for reducible representations. We also derive all irreducible inequivalentQ-representations of a groupG, classifying them according to a suitable generalization of the Wigner-Frobenius-Schur classification.     

Special geometry,cubic polynomials and homogeneous quaternionic spaces

The existing classification of homogeneous quaternionic spaces is not complete. We study these spaces in the context of certainN=2 supergravity theories, where dimensional reduction induces a mapping betweenspecial real, Kähler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the Kähler and quaternionic manifolds in homogeneous holomorphic functions of second degree. We classify all cubic polynomials that have an invariance group that acts transitively on the real manifold. The corresponding Kähler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by Alekseevskiî (and the corresponding special Kähler spaces given by Cecotti), but there is a new class of rank-3 spaces of quaternionic dimension larger than 3. We also point out that some of the rank-4 Alekseevskiî spaces were not fully specified and correspond to a finite variety of inequivalent spaces. A simpler version of the equation that underlies the classification of this paper also emerges in the context ofW ₃ algebras.Communicated by K. Gawedzki     

Quantum mechanics of the quaternionic Hilbert spaces based upon the imprimitivity theorem

We survey the realization of quantum mechanics in quaternionic Hilbert spaces following the methods of Mackey, who examined the complex and real cases exploiting the imprimitivity theorem. We show that there exists a unique unitary skew-adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic imaginary unit. Difficulties in the definition of time reversal, however, arise because of the properties of the quaternionic field. The introduction of an extra imaginary unit, commuting with the others, is suggested in order to implement time reversal properly. In the Appendix we give the proof of the imprimitivity theorem, in the quaternionic case, that we use in the paper.     

Global effects in quaternionic quantum field theory

We present some striking global consequences of a model quaternionic quantum field theory which is locally complex. We show how making the quaternionic structure a dynamical quantity naturally leads to the prediction of cosmic strings and nonbaryonic hot dark matter candidates.     

Quaternionic complexes

Each regular or semi-regular integral affine orbit of the Weyl group of gl(2n + 2, ) invariantly determines a locally exact differential complex on a 4n dimensional quaternionic manifold. This gives quaternionic analogues of Dolbeault cohomology on complex manifolds. We compute the index of such complexes in the hyper-Kähler case, showing that quaternionic cohomology is not trivial.     

Quaternionic electroweak theory and Cabibbo-Kobayashi-Maskawa matrix

We find in our quaternionic version of the electroweak theory an apparently hopeless problem: In going from complex numbers to quaternions, the calculation of the real-valued parameters of the Cabibbo-Kobayashi-Maskawa matrix drastically changes. We aim to explain thisquaternionic puzzle.     

The Map Between Conformal Hypercomplex/ Hyper-Kähler and Quaternionic(-Kähler) Geometry

We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by `ξ-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kähler manifolds is mapped to quaternionic-Kähler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.     

Quaternionic Groups in Physics

We study the left and right action ofquaternionic numbers. The standard problems arising inthe definitions of transpose, determinant, and trace forquaternionic matrices are overcome. We investigate the possibility of formulating a new approach toquaternionic group theory. Our aim is to highlight thepossibility of looking at new quaternionic groups by theuse of left and right operators as fundamental step toward a clear and complete discussion ofunification theories in physics.     

A generalization of the momentum mapping construction for quaternionic Kähler manifolds

We present a method of reduction of any quaternionic Kähler manifold with isometries to another quaternionic Kähler manifold in which the isometries are divided out. Our method is a generalization of the Marsden-Weinstein construction for symplectic manifolds to the non-symplectic geometry of the quaternionic Kähler case. We compare our results with the known construction for Kähler and hyperKähler manifolds. We also discuss the relevance of our results to the physics of supersymmetric non-linear -models and some applications of the method. In particular, we show that the Wolf spaces can be obtained as theU(1) andSU(2) quotients of quaternionic projective spaceH P(n). We also construct an interesting example of compact riemannianV-manifolds(orbifolds) whose metrics are quaternionic Kähler and not symmetric.On leave from the University of Wrocaw, Wrocaw, Poland     

Quaternionic Kähler Detour Complexes and $${\mathcal{N} = 2}$$ Supersymmetric Black Holes

We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional \({{\mathcal N} = 2}\) supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kähler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generate special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston’s complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kähler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry.     

Letter: The Energy Density of the Quaternionic Field as Dark Energy in the Universe

In this article we describe a model of the universe consisting of a mixture of the ordinary matter and a so-called cosmic quaternionic field. The basic idea here consists in an attempt to interpret as the energy density of the quaternionic field whose source is any form of energy including the proper energy density of this field. We set the energy density of this field to and show that the ratio of ordinary dark matter energy density assigned to is constant during the cosmic evolution. We investigate the interaction of the quaternionic field with the ordinary dark matter and show that this field exerts a force on the moving dark matter which might possible create the dark matter in the early universe. Such determined fulfils the requirements asked from the dark energy. In this model of the universe, the cosmological constant, the fine-tuning and the age problems might be solved. Finally, we sketch the evolution of the universe with the cosmic quaternionic field and show that the energy density of the cosmic quaternionic field might be a possible candidate for the dark energy.     

Quaternionic,quaternionic Kähler,and hyper-Kähler supermanifolds

Almost quaternionic, quaternionic, hyper-Kähler, and quaternionic Kähler supermanifolds are introduced and studied.     

Quaternionic Kähler metrics associated with special Kähler manifolds

We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara–Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondence between Kähler manifolds endowed with a Hamiltonian Killing vector field. As an example, we apply this formula in the case of an arbitrary conical Kähler manifold.     

The Spectral Action and Cosmic Topology

The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincaré dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincaré homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces.     

Invariant local twistor calculus for quaternionic structures and related geometries

New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalizations as well as four-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all invariants operators arise from these universal operators and that they may be used to reduce all invariants problems to corresponding algebraic problems involving homomorphisms between modules of certain parabolic subgroups of Lie groups. Explicit application of the operators is illustrated by the construction of all non-standard operators between exterior forms on a large class of the geometries which includes the quaternionic structures.     

Quantum mechanics: From complex to complexified quaternions

This paper is an attempt to simplify and clarify the mathematical language used to express quaternionic quantum mechanics (QQM). In our quaternionic approach the choice of “complex” geometries allows an appropriate definition of momentum operator and gives the possibility to obtain consistent formulations of standard theories. Barred operators represent the key to realizing a set of translation rules between quaternionic and complex quantum mechanics (QM). These translations enable us to obtain a rapid quaternionic counterpart of standard quantum mechanical results.