We consider duality transformations in N = 2, d = 4 Yang-Mills theory coupled to N = 2 supergravity. A symplectic and coordinate covariant framework is established, which allows one to discuss stringy 'classical and quantum duality symmetries' (monodromies), incorporating T and S dualities. In particular, we shall be able to study theories (like N = 2 heterotic strings) which are formulated in symplectic basis where a 'holomorphic prepotential' F does not exist, and yet give general expressions for all relevant physical quantities. Duality transformations and symmetries for the N = 1 matter coupled Yang-Mills supergravity system are also exhibited. The implications of duality symmetry on all N > 2 extended supergravities are briefly mentioned. We finally give the general form of the central charge and the N = 2 semiclassical spectrum of the dyonic BPS saturated states (as it comes by truncation of the N = 4 spectrum).

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We present a geometric construction of the σ model describing type II superstrings propagating on an arbitrary Mtarget. Specializing the covering space of the internal target manifold to be the nine-dimensional group manifold SU(2)3, we discuss the massless vertices both in the (4+9)-dimensional a model and in the D=4 superconformal theory, and show how they are related via dimensional reduction.

We construct infinite families of Lifshitz solutions of D = 10 and D = 11 supergravity with dynamical exponent z = 2. The new solutions are based on five-and seven-dimensional Einstein manifolds and are dual to field theories with Lifshitz scaling in 1+2 and 1+1 spacetime dimensions, respectively. When the Einstein spaces are Sasaki-Einstein, the solutions are supersymmetric.

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