Isotropic Kähler hyperbolic twistor spaces

In this paper we study two natural indefinite almost Hermitian structures on the hyperbolic twistor space of a four-manifold endowed with a neutral metric. We show that only one of these structures can be isotropic Kähler and obtain the precise geometric conditions on the base manifold ensuring this property.     

On the Hyperkähler/Quaternion Kähler Correspondence

A hyperkähler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kähler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkähler/quaternion Kähler correspondence is established.     

Teleparallel Kähler Calculus for Spacetime

In a recent paper [J. G. Vargas and D. G. Torr, Found. Phys. 27, 599 (1997)], we have shown that a subset of the differential invariants that define teleparallel connections in spacetime generates a teleparallel Kaluza-Klein space (KKS) endowed with a very rich Clifford structure. A canonical Dirac equation hidden in this structure might be uncovered with the help of a teleparallel Kähler calculus in KKS. To bridge the gap to such a calculus from the existing Riemannian Kähler calculus in spacetime, we commence the construction of a teleparallel Kähler calculus in spacetime. In the process, we notice: (a) Unknown to him, one of Einstein's equations in his attempt at unification with teleparallelism states that the interior covariant derivative of the torsion is zero. (b) A mechanism exists in the tangent bundle of teleparallel spaces for producing confinement (in the applicable cases, one would have to show why nonconfinement also occurs, rather than the other way around). (c) When the torsion is not zero, the interior covariant derivative in the sense of Kähler, F, does not coincide with *d*F. The system (dF = 0, F = j) rather than (dF = 0, *d*F = j) should then be used for generalizations of Maxwell's electrodynamics.     

Geodesic symmetries and invariant star products on Kähler symmetric spaces

Starting from work by F. A. Berezin, an earlier paper by the author obtained an invariant star product on every nonexceptional symmetric Kähler space. This would be a generalization to those spaces of the star product on ²ⁿ corresponding to Wick quantization. In this Letter we consider, via geometric quantization, the unitary operators corresponding to geodesic symmetries, and we define a Weyl quantization (first defined by Berezin on rank 1 spaces) in a way similar to the way in which the Weyl quantization can be obtained from the Wick quantization on ²ⁿ . We then calculate every Hochschild 2-cochain of another invariant star product, equivalent to the Wick one, which would be a generalization to those spaces of the Moyal star product on ²ⁿ . M. Cahen and S. Gutt have already provided a theorem of existence and essential unicity of an invariant star product on every irreducible Kähler symmetric space.     

From G2 geometry to quaternionic Kähler metrics

We review a construction of quaternionic Kähler metrics starting from a rank 2 distribution in 5 dimensions. We relate it to a more general theory about Einstein deformations of symmetric metrics. Finally we ask some questions on complete metrics and relate them to a Zoll phenomenon.     

Axial anomaly and index theorem for Dirac-Kähler fermions

We present a calculation of the axial anomaly for Dirac-Kähler fermions in two and four dimensions applying the procedure developed by Seeley to the signature operator in the twisted complex. The result is equal to the one for the twisted spin complex times 2^/2 (n: number of dimensions) and agrees with the expressions from the index theorem.     

On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a Kähler or Quaternion-Kähler structure

It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a “lowest” action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for the symmetric space $F_4/{\mathrm{Spin}}_9$, proving that it is valid for all the “possible” holonomies in Berger’s list occurring in that context. The proof is based on a characterization of the first eigenvalue of the Dirac operator given in Milhorat (2005) and Milhorat (2006). By the way, we review an incorrect statement in the proof of the first lemma in Milhorat (2005).     

Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

We consider Lie(G)-valued G-invariant connections on bundles over spaces ${G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}We give a geometric construction of the ${\mathcal{W}_{1+\infty}}We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, \mathbbR×G/H and \mathbbR²×G/H{G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}, where G/H is a compact nearly K?hler six-dimensional homogeneous space, and the manifolds \mathbbR×G/H{\mathbb{R}\times G/H} and \mathbbR²×G/H{\mathbb{R}^2\times G/H} carry G ₂- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on \mathbbR×G/H{\mathbb{R}\times G/H} is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G ₂-structures on \mathbbR×G/H{\mathbb{R}\times G/H}. It is shown that both G ₂-instanton equations can be obtained from a single Spin(7)-instanton equation on \mathbbR²×G/H{\mathbb{R}^2\times G/H}.     

Kähler Moduli Space for a D-Brane at Orbifold Singularities

We develop a method to analyze systematically the configuration space of a D-brane localized at the orbifold singular point of a Calabi–Yau d-fold of the form ℂ ^d /Γ using the theory of toric quotients. This approach elucidates the structure of the K?hler moduli space associated with the problem. As an application, we compute the toric data of the Γ-Hilbert scheme. Received: 9 June 1998 / Accepted: 18 November 1998     

3-Kanal-Impulszähler mit dem Kleinrechner KC 85/3

A 3-channel pulse counter for residence time measuring has been developed on the basis of the micro-computer KC 85/3. This system is characterized by modern conveniences compared with former residence time measuring devices. The technical parameters are comparable with usual radiation measuring assemblies.     

Classical Dynamical r-Matrices¶and Homogeneous Poisson Structures on G/H and K/T

Let G be a finite dimensional simple complex group equipped with the standard Poisson Lie group structure. We show that all G-homogeneous (holomorphic) Poisson structures on G/H, where H⊂G is a Cartan subgroup, come from solutions to the Classical Dynamical Yang–Baxter equations which are classified by Etingof and Varchenko. A similar result holds for a maximal compact subgroup K, and we get a family of K-homogeneous Poisson structures on K/T, where T=K∩H is a maximal torus of K. This family exhausts all K-homogeneous Poisson structures on K/T up to isomorphisms. We study some Poisson geometrical properties of members of this family such as their symplectic leaves, their modular classes, and the moment maps for the T-action. Received: 4 September 1999 / Accepted: 25 January 2000     

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     

A local index theorem for families of\bar \partial -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

We prove a local index theorem for families of \(\bar \partial \) -operators on Riemann surfaces of type (g, n), i.e. of genusg withn>0 punctures. We calculate the first Chern form of the determinant line bundle on the Teichmüller spaceT _g,n endowed with Quillen's metric (where the role of the determinant of the Laplace operators is played by the values of the Selberg zeta function at integer points). The result differs from the case of compact Riemann surfaces by an additional term, which turns out to be the Kähler form of a new Kähler metric on the moduli space of punctured Riemann surfaces. As a corollary of this result we derive, for instance, an analog of Mumford's isomorphism in the case of the universal curve.     

Supersymmetry and the cohomology of (hyper) Kähler manifolds

The cohomology of a compact Kahler (hyperKähler) manifold admits the action of the Lie algebra so(2,1) (so(4,1)). In this paper we show, following an idea of Witten, how this action follows from supersymmetry, in particular from the symmetries of certain supersymmetric sigma models. In addition, many of the fundamental identities in Hodge-Lefschetz theory are also naturally derived from supersymmetry.     

A string calculation of the Kähler potentials for moduli of ZN orbifolds

We present a method for calculating the Kähler potentials of the moduli of Z_N orbifolds directly from string theory. The explicit Kähler potentials associated with b_(1,1) and b_(1,2) moduli are given for any (2,0) symmetric Z_N orbifold. These results are exact at the string tree level.     

Bäcklund transformations for nonlinear sigma models with values in Riemannian symmetric spaces

This work deals with Bäcklund transformations for the principal SL(n, ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Bäcklund transformation for an associated, linearly extended system. Comparison of this multi-Bäcklund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Bäcklund transformations (MBT) is developed, by the introduction of a soliton correlation matrix which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Bäcklund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Bäcklund transformations and their solution for all Riemannian symmetric spaces.Supported in part by the Natural Sciences and Engineering Research Council of Canada, and by the Fonds FCAC pour l'aide et le soutien à la recherche