Some integrable models in quantized spaces

It is shown that in a quantized space determined by the $B_2\quad (O(5)=Sp(4))$ algebra with three dimensional parameters of the length $L^2$, momentum $(Mc)^2$, and action $S$, the spectrum of the Coulomb problem with conserving Runge-Lenz vector coincides with the spectrum found by Schr\"odinger for the space of constant curvature but with the values of the principal quantum number limited from the side of higher values. The same problem is solved for the spectrum of a harmonic oscillator.Comment: 11 pages, LaTe

Updated Analysis of a_1 and a_2 in Hadronic Two-body Decays of B Mesons

Using the recent experimental data of $B\to D^{(*)}(\pi,\rho)$, $B\to D^{(*)} D_s^{(*)}$, $B\to J/\psi K^{(*)}$ and various model calculations on form factors, we re-analyze the effective coefficients a_1 and a_2 and their ratio. QCD and electroweak penguin corrections to a_1 from $B\to D^{(*)}D_s^{(*)}$ and a_2 from $B\to J/\psi K^{(*)}$ are estimated. In addition to the model-dependent determination, the effective coefficient a_1 is also extracted in a model-independent way as the decay modes $B\to D^{(*)}h$ are related by factorization to the measured semileptonic distribution of $B\to D^{(*)}\ell \bar\nu$ at $q^2=m_h^2$. Moreover, this enables us to extract model-independent heavy-to-heavy form factors, for example, $F_0^{BD}(m_\pi^2)=0.66\pm0.06\pm0.05$ and $A_0^{BD^*}(m_\pi^2)=0.56\pm0.03\pm0.04$. The determination of the magnitude of a_2 from $B\to J/\psi K^{(*)}$ depends on the form factors $F_1^{BK}$, $A_{1,2}^{BK^*}$ and $V^{BK^*}$ at $q^2=m^2_{J/\psi}$. By requiring that a_2 be process insensitive (i.e., the value of a_2 extracted from $J/\psi K$ and $J/\psi K^*$ states should be similar), as implied by the factorization hypothesis, we find that $B\to K^{(*)}$ form factors are severely constrained; they respect the relation $F_1^{BK}(m^2_{J/\psi})\approx 1.9 A_1^{BK^*}(m^2_{J/\psi})$. Form factors $A_2^{BK^*}$ and $V^{BK^*}$ at $q^2=m^2_{J/\psi}$ inferred from the measurements of the longitudinal polarization fraction and the P-wave component in $B\to J/\psi K^*$ are obtained. A stringent upper limit on a_2 is derived from the current bound on \ov B^0\to D^0\pi^0 and it is sensitive to final-state interactions.Comment: 33 pages, 2 figures. Typos in Tables I and IX are corrected. To appear in Phys. Rev.

WW--geometry of the Toda systems associated with non-exceptional simple Lie algebras

The present paper describes the $W$--geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the $B,C$ and $D$ series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Pl\"ucker embedding of the $A$-case to the flag manifolds associated with the fundamental representations of $B_n$, $C_n$ and $D_n$, and a direct proof that the corresponding K\"ahler potentials satisfy the system of two--dimensional finite non-periodic (conformal) Toda equations. It is shown that the $W$--geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) of $CP^N$ with appropriate choices of $N$. In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Pl\"ucker embedding. These conditions are automatically fulfiled when Toda equations hold.Comment: 30 pages, no figur

Gravitational anomaly and fundamental forces

I present an argument, based on the topology of the universe, why there are three generations of fermions. The argument implies a preferred gauge group of SU(5), but with SO(10) representations of the fermions. The breaking pattern SU(5) to SU(3)xSU(2)xU(1) is preferred over the pattern SU(5) to SU(4)xU(1). On the basis of the argument one expects an asymmetry in the early universe microwave data, which might have been detected already.Comment: Contribution to the 2nd School and Workshop on Quantum Gravity and Quantum Geometry. Corfu, september 13-20 2009. 10 page

Quantum Dynamics in Non-equilibrium Strongly Correlated Environments

We consider a quantum point contact between two Luttinger liquids coupled to a mechanical system (oscillator). For non-vanishing bias, we find an effective oscillator temperature that depends on the Luttinger parameter. A generalized fluctuation-dissipation relation connects the decoherence and dissipation of the oscillator to the current-voltage characteristics of the device. Via a spectral representation, this result is generalized to arbitrary leads in a weak tunneling regime.Comment: 4 pages, 1 figur

Binding energies and electronic structures of adsorbed titanium chains on carbon nanotubes

We have studied the binding energies and electronic structures of metal (Ti, Al, Au) chains adsorbed on single-wall carbon nanotubes (SWNT) using first principles methods. Our calculations have shown that titanium is much more favored energetically over gold and aluminum to form a continuous chain on a variety of SWNTs. The interaction between titanium and carbon nanotube significantly modifies the electronic structures around Fermi energy for both zigzag and armchair tubes. The delocalized 3d electrons from the titanium chain generate additional states in the band gap regions of the semiconducting tubes, transforming them into metals.Comment: 4 pages, 3 figure

Classification and nondegeneracy of SU(n+1)SU(n+1) Toda system with singular sources

We consider the following Toda system \Delta u_i + \D \sum_{j = 1}^n a_{ij}e^{u_j} = 4\pi\gamma_{i}\delta_{0} \text{in}\mathbb R^2, \int_{\mathbb R^2}e^{u_i} dx -1$,$\delta_0$is Dirac measure at 0, and the coefficients$a_{ij}$form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:$$\sum_{j=1}^n a_{ij}\int_{\R^2}e^{u_j} dx = 4\pi (2+\gamma_i+\gamma_{n+1-i}), \;\;\forall\; 1\leq i \leq n.$$This generalizes the classification result by Jost and Wang for$\gamma_i=0$,$\forall \;1\leq i\leq n$. (ii) We prove that if$\gamma_i+\gamma_{i+1}+...+\gamma_j \notin \mathbb Z$for all$1\leq i\leq j\leq n$, then any solution$u_i$ is \textit{radially symmetric} w.r.t. 0. (iii) We prove that the linearized equation at any solution is \textit{non-degenerate}. These are fundamental results in order to understand the bubbling behavior of the Toda system.Comment: 28 page

Analytic Bethe Ansatz for Fundamental Representations of Yangians

We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian $Y(X_r)$ or its quantum affine analogue $U_q(X^{(1)}_r)$ for $X_r = B_r, C_r$ and $D_r$. Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of $Y(X_r)$. Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the $T$-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.Comment: 45 pages, Plain Te

Dilogarithm Identities in Conformal Field Theory and Group Homology

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all $2 \times 2$ real matrices viewed as a {\it discrete} group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic $K$-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all $2 \times 2$ real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with the summary of a number of open conjectures on the mathematical side.Comment: 20 pages, 2 figures not include

Slowly rotating charged black holes in anti-de Sitter third order Lovelock gravity

In this paper, we study slowly rotating black hole solutions in Lovelock gravity (n=3). These exact slowly rotating black hole solutions are obtained in uncharged and charged cases, respectively. Up to the linear order of the rotating parameter a, the mass, Hawking temperature and entropy of the uncharged black holes get no corrections from rotation. In charged case, we compute magnetic dipole moment and gyromagnetic ratio of the black holes. It is shown that the gyromagnetic ratio keeps invariant after introducing the Gauss-Bonnet and third order Lovelock interactions.Comment: 14 pages, no figur