Deformations of 2k-Einstein structures

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.     

Unique continuation results for Ricci curvature and applications

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss–Codazzi constraint equations under deformations are also discussed.     

Universal deformations of reductive Lie algebras

We construct multiparameter quantizations of reductive Lie algebras which have the property of universality within a certain class of deformations. The universal deformations can be defined so that the algebra structure on each simple component is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear, as a special case of deformations of a semisimple algebra whose simple components remain classical. Deformations are defined as algebras over power series rings and it is essential to require them to be torsion free to secure the universality. The Poincaré-Birkhoff-Witt theorem and the torsion freeness are established for the universal deformation on the basis of results on the representation theory of the deformed algebras.     

Generalized holomorphic structures

We investigate generalized holomorphic structures in generalized complex geometry. We find that a generalized holomorphic vector bundle carries a generalized complex structure on its total space if some additional conditions hold. We prove that generalized holomorphicity is equivalent to the integrability of a distribution on the total space, and a family of linear Dirac structures associated with this distribution is a generalized complex structure if a further condition holds. Under the same condition, we also prove that local generalized holomorphic frames exist around a regular point.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

Universal deformations of simple Lie algebras

This Letter examines the question of the structure of the Hopf algebra deformations of the universal enveloping algebras of the simple Lie algebras. Deformations of a complex algebra A are viewed as algebras defined over formal power series rings that specialize to A when the parameters go to 0. Only the case of U(sl(2,C)) is treated but the methods are general. Under the Ansatz that the two Borel subalgebras are deformed as Hopf algebras but possibly differently, we construct a universal two-parameter deformation.     

On complex Landsberg and Berwald spaces

In this paper, we study complex Landsberg spaces and some of their important subclasses. The tools of this study are the Chern-Finsler, Berwald, and Rund complex linear connections. We introduce and characterize the class of generalized Berwald and complex Landsberg spaces. The intersection of these spaces gives the so-called G-Landsberg class. This last class contains two other kinds of complex Finsler spaces: strong Landsberg and G-Kähler spaces. We prove that the class of G-Kähler spaces coincides with complex Berwald spaces, in Aikou’s (1996) [1] sense, and it is a subclass of the strong Landsberg spaces. Some special complex Finsler spaces with (α,β)-metrics offer examples of generalized Berwald spaces. Complex Randers spaces with generalized Berwald and weakly Kähler properties are complex Berwald spaces.     

Poisson Lie groups and pentagonal transformations

Symplectic pentagonal transformations are intimately related to global versions of Poisson Lie groups (Manin groups, S ^*-groups, or symplectic pseudogroups). Symplectic pentagonal transformations of cotangent bundles, preserving the natural polarization, are shown to be in one to one correspondence with pentagonal transformations of the base manifold with a cocycle (if the base is connected and simply connected). By the results of Baaj and Skandalis, this allows to quantize (at the C ^*-algebra level!) those Poisson Lie groups, whose associated symplectic pentagonal transformation admits an invariant polarization. The (2n)²-parameter family of Poisson deformations of the (2n+1)-dimensional Heisenberg group described by Szymczak and Zakrzewski is shown to fall into this case.Supported by Alexander von Humboldt Foundation. On leave from Department of Mathematical methods in Physics, Warsaw University, Poland.     

Relativistic point interaction

A four-parameter family of all self-adjoint operators corresponding to the one-dimensional Dirac Hamiltonian with point interaction is characterized in terms of boundary conditions. The spectrum and eigenvectors, and the scattering parameters are calculated. It is shown that the nonrelativistic limit reproduces (in the norm resolvent sense) the four-parameter family of Schrödinger operators with point interaction, their eigenvalues and scattering parameters.Graduiertenkolleg Geometrie und mathematische Physik.Alexander von Humboldt fellow. On leave of absence from SISSA, Trieste, Italy.     

Linear Perturbations of Hyperkähler Metrics

We study general linear perturbations of a class of 4d real-dimensional hyperkähler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d + 1 variables, as opposed to the functions of d + 1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor space. The deformed Kähler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah–Hitchin manifold away from its negative mass Taub-NUT limit.     

The concept of a quantum semisimple group

We explore the possibility of introducing the concept of a quantum semisimple group by exhibiting a class of deformations of classical groups whose Dynkin diagrams are disconnected at the classical level, but become connected at the quantum level. The possibility of applications to the quantization of Lorentz and Poincaré groups are mentioned.     

Deformation of LeBrun’s ALE Metrics with Negative Mass

In this article we investigate deformations of a scalar-flat Kähler metric on the total space of complex line bundles over ${\mathbb{CP}^1}$ constructed by C. LeBrun. In particular, we find that the metric is included in a one-dimensional family of such metrics on the four-manifold, where the complex structure in the deformation is not the standard one.     

Covariantising the Beltrami equation in W-gravity

Recently, certain higher-dimensional complex manifolds were obtained by S. Govindarajan [1] by associating a higher dimensional uniformisation to the generalised Teichmüller spaces of Hitchin. The extra dimensions are provided by the times of the generalised KdV hierarchy. In this Letter, we complete the proof that these manifolds provide the analog of superspace for W-gravity and that W-symmetry linearises on these spaces. This is done by explicitly constructing the relationship between the Beltrami differentials which naturally occur in the higher-dimensional manifolds and the Beltrami differentials which occur in W-gravity. This also resolves an old puzzle regarding the relationship between KdV flow. and W-diffeomorphisms.Dedicated to the memory of Claude Itzykson.     

A note on the super Krichever map

We consider the geometrical aspects of the Krichever map in the context of Jacobian super KP hierarchy. We use the representation of the hierarchy based on the Faà di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian.     

Eisenstein type series for Calabi–Yau varieties

In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi–Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the q-expansion form and the Yukawa coupling turns out to be rational in these functions. We prove that these functions are algebraically independent over the field of complex numbers, and hence, the algebra generated by such functions can be interpreted as the theory of (quasi) modular forms attached to the one parameter family of Calabi–Yau varieties. Our result is a reformulation and realization of a problem of Griffiths around seventies on the existence of automorphic functions for the moduli of polarized Hodge structures. It is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.     

Effects of interstitial oxygen and nitrogen on the mechanical properties of niobium-titanium alloys

The effects were studied of the clustering of substitutional titanium with interstitial oxygen or nitrogen on the strength of the niobium −1 at.-% titanium alloy. The observed values of the yield parameters and flow stress were resolved into a sum of contributions due to the solution strengthening and intrinsic strength components. From the obtained relation the substitutional-interstitial complex hardening coefficients for oxygen and nitrogen in the alloy were determined and compared with the respective solution hardening coefficients for unalloyed niobium. It was also found that the strengthening contribution due to atomic size mismatch and modulus differences introduced by unassociated titanium is small in comparison with that due to substitutional-interstitial complexing. In addition, the alloy displays both higher strength and elongation with respect to the base material. Strain-ageing experiments were carried out on the same alloy and the results were combined with the information obtained from the stress-strain tests and the internal friction measurements reported in a previous work.