The Semiclassical Limit for SU(2)SU(2) and SO(3)SO(3) Gauge Theory on the Torus

We prove that for $SU(2)$ and $SO(3)$ quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure d\mu_T(\o) =N_T^{-1}e^{-S_{YM}(\o)/T}[{\cal D}\o] converge, as $T\downarrow 0$, to integrals with respect to a symplectic volume measure $\mu_0$ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly.Comment: 18 page

A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric. The class of the nearly Kaehler manifolds with respect to the first almost complex structure is of special interest. It is proved that D has a D-parallel torsion and is weak if it is not flat. Some curvature properties of these manifolds are studied.Comment: 18 page