We have seen that irreducible finite dimensional representations of a complex simple Lie algebra g or corresponding compact Lie group are classified and can be constructed starting from an integral dominant weight. The dominance condition depends upon a choice of positive roots (or equivalently, a choice of invariant complex structure on the flag manifold.) An obvious question is that of what h...

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, in...

We consider S 2 bundles P and P ′ of totally null planes of maximal dimension over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP ′ of P and P ′ is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical obje...

For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z , describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations ofA. In particular, this gives rise to a gauge symmetry described by the action of Z . A distinguis...

The Weyl algebra A of continuous functions and exponentiated fluxes, introduced by Ashtekar, Lewandowski and others, in quantum geometry is studied. It is shown that, in the piecewise analytic category, every regular representation of A having a cyclic and diffeomorphism invariant vector, is already unitarily equivalent to the fundamental representation. Additional assumptions concern the dimen...

We are interested in remainder estimates in the Weyl formula for the asymptotic number of eigenvalues of certain elliptic operators on R and on a smooth compact manifold without boundary. The main aim of this paper is to compare spectral asymptotics of operators with irregular coefficients and certain classes of smoothed operators for which the Weyl formula is derived by means of elementary pse...

We calculate the Weyl anomaly for conformal field theories that can be described via the adS/CFT correspondence. This entails regularizing the gravitational part of the corresponding supergravity action in a manner consistent with general covariance. Up to a constant, the anomaly only depends on the dimension d of the manifold on which the conformal field theory is defined. We present concrete ...

Let X be a complex manifold with strongly pseudoconvex boundary M . If ψ is a defining function for M , then − logψ is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form σ = i∂∂(− logψ) is a symplectic structure on the complement of M in a neighborhood in X of M ; it blows up along M . The Poisson structure obtained by inverting σ extends smoothly across M and determines a cont...

For a positive integer c, letHc be the subgroup of G obtained when x, y, cz are integers. The Heisenberg manifold Mc is the quotient G/Hc. Nonzero Poisson brackets on Mc invariant under left translation by G are parametrized by two real parameters μ, ν with μ + ν 6= 0 [4]. For each positive integer c and real numbers μ, ν Rieffel constructed a C*-algebra A μ,ν , quantum Heisenberg manifold (QHM...

Since the fundamental work of Bayen et al. [1] in the seventies, a lot of effort has been dedicated to show the existence of deformations of a Poisson manifold. Some landmarks in this way were the proof of the existence of differential star products for symplectic manifolds which was done independently by De Wilde and Lecomte [2] and Fedosov [5], using different constructions. It turned out tha...