Hamiltonian circle actions on symplectic manifolds and the signature

Let M be a symplectic manifold with a Hamiltonian circle action with isolated fixed points. We prove that σ (M) = b₀(M) − b₂(M) + b₄(M) − b₆(M) + … where σ (M) is the signature of M and b_i(M) is the ith Betti number of M.     

Exact symplectic structures and a classical model for the Dirac electron

We show how the classical model for the Dirac electron of Barut and coworkers can be obtained as a Hamiltonian theory by constructing an exact symplectic form on the total space of the spin bundle over spacetime.     

Quantization of K?hler manifolds. IV

We use Berezin's dequantization procedure to define a formal *-product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.This work was partially supported by EC contract CHRX-CT92-0050.     

Topological charges in monopole theories

Let us consider a monopole theory with a compact, simply connected gauge group and the Higgs field in the adjoint representation. Using root theory we show that.(i) The homotopy class of the Higgs field is ap-tuple of integers wherep is the dimension of the centre of the residual symmetry group. These Higgs charges can be expressed as surface integrals of differential forms.(ii) To any invariant polynomial on the Lie algebra is associated a topological invariant which turns out to be a combination of the Higgs charges.(iii) Electric charge is quantized. The monopole's magnetic charge is a combination — with the Higgs charges as coefficients — ofp basic magnetic charges which satisfy generalized Dirac conditions.The example ofG=SU(N) is worked out in detail.     

Singular unitary representations and indefinite harmonic theory

Square-integrable harmonic spaces are defined and studied in a homogeneous indefinite metric setting. In the process, Dolbeault cohomologies are unitarized, and singlar unitary representations are obtained and studied.