Noncommutative Geometry and Geometric Quantization

title: Fat bundles and symplectic manifolds
abstract: A fat bundle is, roughly speaking, a bundle which has a lot of curvature, as opposed to a flat bundle, which has none. (The precise definition is given in Section 1.) For example, a bundle with structure group S’ is fat if it has a connection whose curvature is a symplectic structure on the base. The notion of fat bundle arises naturally from the study of riemannian fibre bundles with totally geodesic fibres. Positivity of certain sectional curvatures turns out to be equivalent to fatness of the connection determined by the orthogonal complements to the fibres. It was for this reason that fat bundles were introduced in a manuscript written at MIT in 1968 and circulated informally [20]. (The term unj7at used in that manuscript to mean fat is rather lacking in euphony; it is also slightly misleading, since a bundle which is not flat might not be unflat either-most bundles are somewhere in between.) Some of the results in this paper are taken from the 1968 manuscript, in particular Lemma 3.1, Theorem 4.1 and its corollaries, Theorem 6.1, and Theorem 7.2, but there are substantial additions as well. Using an idea of Sternberg, we show in Section 3 how to construct symplectic manifolds out of any fat bundle, using coadjoint orbits of the structure group; we then continue the discussion by using an idea of Thurston [ 181 to show that, whatever the connection, it is often possible to put a symplectic structure on the total space of a bundle whose tibres and base are symplectic. (In particular, it is sufficient that the bundle be compact and have simply connected fibres.) A problem which was posed and partially solved in [20] was to determine the fat 3-sphere bundles over S4. This problem has just been solved by Derdziriski and Rigas [7]-there is only the Hopf tibration S3 + S’ + S4. Their solution (see Section 7) uses ideas generated by the recent revival of interest in 4-dimensional riemarmian geometry  centered around the notion of self-duality. Several important contributions to the theory of fat bundles have been

title: GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY
abstract: When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

title: Lectures on the geometry of quantization

title: The Differential Geometry of Fedosov’s Quantization
abstract: 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.

title: Courant Algebroids and Strongly Homotopy Lie Algebras
abstract: 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.

title: Tau Electroweak Couplings
abstract: We review world-averagemeasurements of the tau lepton electroweak couplings, in both decay (including Michel parameters) and in production (Z 0 ! + and W ! ). We review the searches for anomalous weak and EM dipole couplings. Finally, we present the status of several other tau lepton studies: searches for lepton flavor violating decays, neutrino oscillations, and tau neutrino mass limits.

title: The search for millicharged particles at SLAC
abstract: Quantization of electric charge has been probed to a high degree of precision using a variety of experimental techniques. However, the theoretical underpinning of charge quantization has yet to be determined, leaving open the possibility that quantization is not a fundamental law of nature, and that free particles with fractional electric charge may one day be encountered. This dissertation describes a search for particles with very small (10{sup -3}e or below) electric charge, carried out at an electron accelerator (SLAC). The particles sought were assumed to be long lived or stable and to participate primarily in electromagnetic interactions. The search relied upon a large (1.3-meter) scintillation counter to detect low levels of ionization generated by relativistic fractionally charged particles produced in the collision of a 29.5-GeV electron beam with a six radiation length tungsten target. The data reveal no evidence of fractional charge, and allow for the exclusion of a portion of charge-mass parameter space spanning roughly 10{sup -7} to 10{sup 2} MeV/c{sup 2} and 10{sup -2} to 10{sup -5} e.

title: Recent commissioning experience on the SLC ARCS
abstract: The Arc transport line, which brings high-energy, high-intensity electron and positron bunches from the SLAC ( linac to the Stanford Linear Collider (SLC) final focus section, has been in operation for the past few years. The techniques developed for the optical tuneup and diagnostics, performance, and ongoing improvement programs are reviewed. In the operation of the SLC Arcs, powerful tools are implemented to: (i) correct systematic gradient errors due mainly to horizontal placement errors (phasefix), (ii) correct systematic skew-field errors due mainly to vertical placement errors (skewfix), and (iii) semiempirically modify the effective beta functions at the Arc exit (wirefix).<<ETX>>

title: Search for neutrinoless tau decays: tau ---> e gamma and tau ---> mu gamma
abstract: A search for the lepton family number violating decays {tau} {yields} e{gamma} and {tau} {yields} {mu}{gamma} has been performed using CLEO-II data. No evidence of a signal has been found and the corresponding upper limits are B({tau} {yields} e{gamma}) < 2.7 x 10{sup -6} and B({tau} {yields} {mu}{gamma}) < 3.0 c 10{sup -6} at 90% CL.

title: Measurement of hadronic spectral moments in tau decays and a determination of alpha-s