On a conjecture of Guillemin and Sternberg in geometric quantization of multiplicity-free symplectic spaces

We study the relation between multiplicity-free symplectic manifolds and the multiplicities of group representations obtained by geometric quantization. With the help of a general equivalence theorem we can prove a conjecture of Guillemin and Sternberg in the compact Kähler case.     

Symplectic and hyperkähler structures in a dimensional reduction of the Seiberg-Witten equations with a Higgs field

In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and we study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g ≥ 1, gives rise to a family of moduli spaces carrying a hyperkähler structure. For the full set of equations the corresponding moduli space does not have the aforementioned hyperkähler structure but has a natural symplectic structure. For the case of the torus, g = 1, we show that the full set of equations has a solution, different from the “vortex solutions”.