15823008
Classical Theta Functions and Quantum Tori
The Schwartz kernel of the multiplication operation on a quantum torus is shown to be the distributional boundary value of a classical multivariate theta function. The kernel satisfies a Schrodinger equation in which the role of time is played by the deformation parameter 11 and the role of the hamiltonian by a Poisson structure. At least in some special cases, the kernel can be written as a sum of products of single variable theta functions. A quantum torus is, by definition, a "space" whose (noncommutative) algebra of functions is obtained by deformation of the (commutative) algebra of functions on a torus (R/Zy. In this note, we observe that the Schwartz kernels of the multiplication operators for a class of quantum tori are the (distributional) boundary values of classical theta functions, and we draw some conclusions from this observation. In particular, we show that each of these kernels satisfies a Schrodinger equation, in which the role of time is played by the deformation parameter h, and the role of the hamiltonian is played by the Poisson structure associated with the deformation. Thus, the "evolution in Plank's constant" of a torus is like the time evolution of a free particle, with the initial state (the Schwartz kernel of the operator of pointwise multiplication) being a delta function, like the initial state of a particle with certain position and totally uncertain momentum. This evolution equation suggests a possible answer to the question of how a deformed product must behave after the deformation parameter h has left the first infinitesimal neighborhood of zero. We note that our work seems to be quite different from Manin's study [2] of quantum theta functions, which are special functions on the quantum tori themselves. Communicated by T. Kawai, September 13, 1993. 1991 Mathematics Subject Classifications : 81S10, 33D10, 58B30. Department of Mathematics, University of California, Berkeley, CA 94720 USA. Research partially supported by NSF Grant DMS-93-09653.
17756587
Poisson geometry with a 3 form background
We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable Courant algebroids. The additive group of 2-forms acts on twisted Poisson structures and permits them to be seen as glued from ordinary Poisson structures defined on local patches. We conclude with remarks on deformation quantization and twisted symplectic groupoids.
15329636
Hopfish structure and modules over irrational rotation algebras
Inspired by the group structure on $S^1/ \bbZ$, we introduce a weak hopfish structure on an irrational rotation algebra $A$ of finite Fourier series. We consider a class of simple $A$-modules defined by invertible elements, and we compute the tensor product between these modules defined by the hopfish structure. This class of simple modules turns out to generate an interesting commutative unital ring.
16221419
The Differential Geometry of Fedosov’s Quantization
B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov’s operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally, some remarks are made on the implications for deformation quantization of Fedosov’s index theorem on general symplectic manifolds.
17256168
Courant Algebroids and Strongly Homotopy Lie Algebras
Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the direct sum of tangent and cotangent bundles with the bracket introduced by T. Courant for the study of Dirac structures. Within the category of Courant algebroids one can construct the doubles of Lie bialgebroids, the infinitesimal objects for Poisson groupoids. We show that Courant algebroids can be considered as strongly homotopy Lie algebras.
121794904
A universal phase space for particles in Yang-Mills fields
Given a principal G-bundle P over M and a Hamiltonian G-space Q, one may construct the reduced symplectic manifold (T*P x Q)0. When a connection on P is chosen, this manifold becomes a bundle over T*M with fibre Q. It is shown that this bundle is precisely the phase space constructed by Sternberg for a classical particle in a Yang-Mills field.
209881353
Measurement of the time dependence of B0 - anti-B0 oscillations using inclusive dilepton events
A preliminary study of time dependence of B^0-anti-B^0 oscillations using dilepton events is presented. The flavor of the B meson is determined by the charge sign of the lepton. To separate signal leptons from cascade and fake leptons we have used a method which combines several discriminating variables in a neural network. The time evolution of the oscillations is studied by reconstructing the time difference between the decays of the B mesons produced by the Y(4S) decay. With an integrated luminosity of 7.7 fb-1 collected on resonance by BABAR at the PEP-II asymmetric B Factory, we measure the difference in mass of the neutral B eigenstates, Delta_mB0, to be (0.507+/-0.015+/-0.022) x 10^{12} hbar-s^{-1}.
2992264
Status of tau physics
We review, very briefly, many of the important results arising in the study of tau lepton production and decay, with emphasis on recent results. We discuss the studies of topological and exclusive branching fractions, inclusive semi-leptonic tau decay spectral functions, extraction of QCD parameters including α_S and the strange quark mass, exclusive semileptonic decays and low energy meson dynamics, and rare and forbidden (neutrinoless) decays. Special attention is paid to recent searches for CP non-conservation in tau pair production and tau decay.
125934668
Study of CP violating asymmetries in $B \to \pi^\pm \pi^\mp$, $K^\pm \pi^\mp$ decays
2236887
The VERITAS Trigger System
The VERITAS gamma-ray observatory, situated in southern Arizona, is an array of four 12m diameter imaging Cherenkov telescopes, each with a 499-pixel photomultiplier-tube camera. The instrument is designed to detect astrophysical gamma rays at energies above 100 GeV. At the low end of the VERITAS energy range, fluctuations in the night sky background light and single muons from cosmic-ray showers constitute significant backgrounds. VERITAS employs a three-tier trigger system to reduce the rate of these background events: an initial trigger which acts at the single pixel level, a pattern trigger which acts on the relative timing and pixel level, a pattern trigger which acts on the relative timing and distribution of pixel-level triggers within a single telescope camera, and an array-level trigger which requires simultaneous observation of an air-shower event in multiple telescopes. This final coincidence requirement significantly reduces the rate of background events, particularly those due to single muons. In this paper, the implementation of all levels of the VERITAS trigger system is discussed and their joint performance is characterized.
121974514
A measurement of CP-violation parameters in B0 anti-B0 mixing using partially reconstructed D*- l+ nu/l events at BaBar
CP violation in B0B0bar mixing is characterized by the value of the parameter |q/p| being different from 1, and the Standard Model predicts this difference to be smaller than 10^{-3}. We present a measurement of this parameter using a partial reconstruction of one of the B mesons in the semileptonic channel D*- l+ nu_l, where only the hard lepton and the soft pion from the D*- --> D0bar pi- decay are reconstructed. The flavor of the other B is determined by means of lepton tagging. The determination of |q/p| is then performed with a fit to the proper time difference of the two B decays. We use a luminosity of 200.8 fb^-1, collected at the Y(4S) resonance by the BaBar detector at the PEP-II asymmetrical-energy e+e- collider, in the period 1999-2004. We obtain the preliminary result: |q/p| - 1 = (6.5 +/- 3.4(stat.) +/- 2.0(syst.)) x 10^{-3}