Holomorphic Cartan geometries and Calabi–Yau manifolds

Let M$M$ be a connected complex projective manifold such that _c1(T^(1,0)M)=0$c_1\left(T^{(1,0)}M\right)=0$. If M$M$ admits a holomorphic Cartan geometry, then we show that M$M$ is holomorphically covered by an abelian variety.     

Resolutions of non-regular Ricci-flat Kähler cones

We present explicit constructions of complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki–Einstein manifolds. The metrics are constructed from a complete Kähler–Einstein manifold (V,_gV)$(V,g_V)$ of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kähler metrics on the total spaces of (i) holomorphic C²/_Zp${\mathbb{C}}^2/{\mathbb{Z}}_p$ orbifold fibrations over V$V$, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP¹${\mathbb{W}\mathbb{C}\mathbb{P}}^1$, with generic fibres being the canonical complex cone over V$V$, and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kähler metrics on the total spaces of (a) rank two holomorphic vector bundles over V$V$, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base V$V$. When V=CP¹$V={\mathbb{C}\mathbb{P}}^1$ our results give Ricci-flat Kähler orbifold metrics on various toric partial resolutions of the cone over the Sasaki–Einstein manifolds Y^p,q$Y^{p,q}$.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

Pseudo-Hermitian generalized Dirac oscillators

We study generalized Dirac oscillators with complex interactions in (1+1)$(1+1)$ dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are η$\eta$-pseudo-Hermitian with respect to certain metric operators η$\eta$. Exact solutions for the generalized Dirac oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with an anti-Jaynes–Cummings-type model and by spin flipping they can also be identified with a Jaynes–Cummings-type model.     

On the geometry of isomonodromic deformations

This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs (V,∇)$(V,\nabla )$ of vector bundles and connections as being obtained by “twists” supported over points of a fixed vector bundle _V0$V_0$ with a fixed connection _∇0${\nabla }_0$; this gives two deformations, one, isomonodromic, of (V,∇)$(V,\nabla )$, and another induced from the isomonodromic deformation of (_V0,_∇0)$(V_0,{\nabla }_0)$. The difference between the two will be Hamiltonian.     

The basic bundle gerbe on unitary groups

We consider the construction of the basic bundle gerbe on SU(n)$SU\left(n\right)$ introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n)$U\left(n\right)$ and _Up(H)$U_p\left(H\right)$, the Banach Lie group of unitaries differing from the identity by an element of a Schatten ideal. In all these cases we give an explicit connection and curving on the basic bundle gerbe and calculate the real Dixmier–Douady class. Extensive use is made of the holomorphic functional calculus for operators on a Hilbert space.     

Goldman flows on the Jacobian

We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1)$U\left(1\right)$, which identifies with the Jacobian.     

Massive 3-loop ladder diagrams for quarkonic local operator matrix elements

3-loop diagrams of the ladder-type, which emerge for local quarkonic twist-2 operator matrix elements, are computed directly for general values of the Mellin variable N using Appell-function representations and applying modern summation technologies provided by the package Sigma and the method of hyperlogarithms. In some of the diagrams generalized harmonic sums with ξ∈{1,1/2,2}$\xi \in \{1,1/2,2\}$ emerge beyond the usual nested harmonic sums. As the asymptotic representation of the corresponding integrals shows, the generalized sums conspire giving well behaved expressions for large values of N  . These diagrams contribute to the 3-loop heavy flavor Wilson coefficients of the structure functions in deep-inelastic scattering in the region Q²?m²$Q^2?m^2$.     

Classification of Lie algebras with naturally graded quasi-filiform nilradicals

The whole class of complex Lie algebras g$\mathfrak{g}$ having a naturally graded nilradical with characteristic sequence c(g)=(dimg−2,1,1)$c\left(\mathfrak{g}\right)=(dim\mathfrak{g}-2,1,1)$ is classified. It is shown that up to one exception, such Lie algebras are solvable.     

On the scaling ranges of detrended fluctuation analysis for long-term memory correlated short series of data

We examine the scaling regime for the detrended fluctuation analysis (DFA)—the most popular method used to detect the presence of long-term memory in data and the fractal structure of time series. First, the scaling range for DFA is studied for uncorrelated data as a function of time series length L$L$ and the correlation coefficient of the linear regression R²$R^2$ at various confidence levels. Next, a similar analysis for artificial short series of data with long-term memory is performed. In both cases the scaling range λ$\lambda$ is found to change linearly—both with L$L$ and R²$R^2$. We show how this dependence can be generalized to a simple unified model describing the relation λ=λ(L,R²,H)$\lambda =\lambda (L,R^2,H)$ where H$H$ (1/2≤H≤1$1/2\le H\le 1$) stands for the Hurst exponent of the long range autocorrelated signal. Our findings should be useful in all applications of DFA technique, particularly for instantaneous (local) DFA where a huge number of short time series has to be analyzed at the same time, without possibility of checking the scaling range in each of them separately.