Geometric quantization of mechanical systems with time-dependent parameters

Quantum systems with adiabatic classical parameters are widely studied, e.g., in the modern holonomic quantum computation. We here provide complete geometric quantization of a Hamiltonian system with time-dependent parameters, without the adiabatic assumption. A Hamiltonian of such a system is affine in the temporal derivative of parameter functions. This leads to the geometric Berry factor phenomena.Comment: 20 page

Tri-hamiltonian vector fields, spectral curves and separation coordinates

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P_0, P_1, P_2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P_1 - \lambda P_0) and (P_2 - \mu P_0); (ii) a suitable set of vector fields, preserving P_0 but transversal to its symplectic leaves. The frameworks is applied to Lax equations with spectral parameter, for which not only it unifies the separation techniques of Sklyanin and of Magri, but also provides a more efficient ``inverse'' procedure not involving the extraction of roots.Comment: 49 pages Section on reduction revisite

Mobility of stretched water

To study the mobility of stretched SPC/E water and its dependence on temperature and density, five molecular dynamics computer simulation runs were performed. Three runs were performed at temperature 300 K and densities 1.0, 0.9, and 0.8 g/cc. Two more runs were performed at temperature 273 K and densities 1.0 and 0.9 g/cc. At temperature 300 K, the translational diffusion coefficient of the stretched SPC/E water increased with the stretch, at temperature 273 K the translational diffusion decreased with the stretch. This behavior is correlated with the observed changes in the hydrogen bonding pattern of water.To study the mobility of stretched SPC/E water and its dependence on temperature and density, five molecular dynamics computer simulation runs were performed. Three runs were performed at temperature 300 K and densities 1.0, 0.9, and 0.8 g/cc. Two more runs were performed at temperature 273 K and densities 1.0 and 0.9 g/cc. At temperature 300 K, the translational diffusion coefficient of the stretched SPC/E water increased with the stretch, at temperature 273 K the translational diffusion decreased with the stretch. This behavior is correlated with the observed changes in the hydrogen bonding pattern of water

Lipoprotein lipase regulates hematopoietic stem progenitor cell maintenance through DHA supply.

Lipoprotein lipase (LPL) mediates hydrolysis of triglycerides (TGs) to supply free fatty acids (FFAs) to tissues. Here, we show that LPL activity is also required for hematopoietic stem progenitor cell (HSPC) maintenance. Knockout of Lpl or its obligatory cofactor Apoc2 results in significantly reduced HSPC expansion during definitive hematopoiesis in zebrafish. A human APOC2 mimetic peptide or the human very low-density lipoprotein, which carries APOC2, rescues the phenotype in apoc2 but not in lpl mutant zebrafish. Creating parabiotic apoc2 and lpl mutant zebrafish rescues the hematopoietic defect in both. Docosahexaenoic acid (DHA) is identified as an important factor in HSPC expansion. FFA-DHA, but not TG-DHA, rescues the HSPC defects in apoc2 and lpl mutant zebrafish. Reduced blood cell counts are also observed in Apoc2 mutant mice at the time of weaning. These results indicate that LPL-mediated release of the essential fatty acid DHA regulates HSPC expansion and definitive hematopoiesis

A class of Poisson-Nijenhuis structures on a tangent bundle

Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis structure from a given type (1,1) tensor field J on Q. It is argued that the complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to co-tangent manifold of Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case that Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.Comment: 22 page

The general (2,2) gauged sigma model with three--form flux

We find the conditions under which a Riemannian manifold equipped with a closed three-form and a vector field define an on--shell N=(2,2) supersymmetric gauged sigma model. The conditions are that the manifold admits a twisted generalized Kaehler structure, that the vector field preserves this structure, and that a so--called generalized moment map exists for it. By a theorem in generalized complex geometry, these conditions imply that the quotient is again a twisted generalized Kaehler manifold; this is in perfect agreement with expectations from the renormalization group flow. This method can produce new N=(2,2) models with NS flux, extending the usual Kaehler quotient construction based on Kaehler gauged sigma models.Comment: 24 pages. v2: typos fixed, other minor correction

A sigma model field theoretic realization of Hitchin's generalized complex geometry

We present a sigma model field theoretic realization of Hitchin's generalized complex geometry, which recently has been shown to be relevant in compactifications of superstring theory with fluxes. Hitchin sigma model is closely related to the well known Poisson sigma model, of which it has the same field content. The construction shows a remarkable correspondence between the (twisted) integrability conditions of generalized almost complex structures and the restrictions on target space geometry implied by the Batalin--Vilkovisky classical master equation. Further, the (twisted) classical Batalin--Vilkovisky cohomology is related non trivially to a generalized Dolbeault cohomology.Comment: 28 pages, Plain TeX, no figures, requires AMS font files AMSSYM.DEF and amssym.tex. Typos in eq. 6.19 and some spelling correcte

Cohomology of skew-holomorphic Lie algebroids

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.Comment: 16 pages. v2: Final version to be published in Theor. Math. Phys. (incorporates only very minor changes

Jacobi-Nijenhuis algebroids and their modular classes

Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids

Leibniz 2-algebras and twisted Courant algebroids

In this paper, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac structures of twisted Courant algebroids give rise to 2-term $L_\infty$-algebras and geometric structures behind them are exactly $H$-twisted Lie algebroids introduced in \cite{Grutzmann}.Comment: 22 pages, to appear in Comm. Algebr