Following hep-th/0211098 we study the matrix model which describes the topological A-model on T * (S 3 /Z p ). We show that the resolvent has square root branch cuts and it follows that this is a p cut single matrix model. We solve for the resolvent and find the spectral curve. We comment on how this is related to large N transitions and mirror symmetry.

We apply an analytic description to the inclusive decay of the τ lepton. We argue that this method gives not only a self-consistent description of the process both in the timelike region by using the initial expression for R τ and in the spacelike domain by using the analytic properties of the hadronic correlator, but also leads to the fact that theoretical uncertainties associated with unknown higher-loop contributions and renormalization scheme dependence can be reduced dramatically.

We demonsrate that the spectral curve of the matrix model for Chern-Simons theory on the Lens space S 3 /Z p is precisely the Riemann surface which appears in the mirror to the blownup, orbifolded conifold. This provides the first check of the A-model large

For matrix models with measure on the Lie algebra of SO/Sp, the sub-leading free energy is given by F 1 (S) = ± 1 4 ∂F 0 (S) ∂S . Motivated by the fact that this relationship does not hold for Chern-Simons theory on S 3 , we calculate the sub-leading free energy in the matrix model for this theory, which is a Gaussian matrix model with Haar measure on the group SO/Sp. We derive a quantum loop equation for this matrix model and then find that F 1 is an integral of the leading order resolvent over the spectral curve.We explicitly calculate this integral for quadratic potential and find agreement with previous studies of SO/Sp Chern-Simons theory.

The structure of the reduced phase space arising in the Hamiltonian reduction of the phase space corresponding to a free particle motion on the group SL(2, R) is investigated. The considered reduction is based on the constraints similar to those used in the Hamiltonian reduction of the Wess-Zumino-Novikov-Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to the union of two two-dimensional planes, or to the cylinder S 1 × R. Canonical coordinates are constructed for the both cases, and it is shown that in the first case the reduced phase space is symplectomorphic to the union of two cotangent bundles T * (R) endowed with the canonical symplectic structure, while in the second case it is symplectomorphic to the cotangent bundle T * (S 1 ) also endowed with the canonical symplectic structure.