We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection∇ with torsion H, the NS⊗NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing spinors of the null backgrounds have stability subgroups K ⋉ R 8 in Spin(9, 1), for K = Spin(7), SU (4), Sp(2), SU (2)×SU (2) and {1}, and the Killing spinors of the timelike backgrounds have stability subgroups G 2 , SU (3), SU (2) and {1}. The former admit a single null∇-parallel vector field while the latter admit a timelike and two, three, five and nine spacelike∇-parallel vector fields, respectively. The spacetime of the null backgrounds is a Lorentzian two-parameter family of Riemannian manifolds B with skew-symmetric torsion. If the rotation of the null vector field vanishes, the holonomy of the connection with torsion of B is contained in K. The spacetime of time-like backgrounds is a principal bundle P with fibre a Lorentzian Lie group and base space a suitable Riemannian manifold with skew-symmetric torsion. The principal bundle is equipped with a connection λ which determines the non-horizontal part of the spacetime metric and of H. The curvature of λ takes values in an appropriate Lie algebra constructed from that of K. In addition dH has only horizontal components and contains the Pontrjagin class of P . We have computed in all cases the Killing spinor bilinears, expressed the fluxes in terms of the geometry and determine the field equations that are implied by the Killing spinor equations.

Abstract:In this paper we construct Birkhoff coordinates for the focusing nonlinear Schrödinger equation near the zero solution.

We describe the geometry of all type II common sector backgrounds with two supersymmetries. In particular, we determine the spacetime geometry of those supersymmetric backgrounds for which each copy of the Killing spinor equations admits a Killing spinor. The stability subgroups of both Killing spinors are Spin(7) ⋉ R 8 , SU (4) ⋉ R 8 and G 2 for IIB backgrounds, and Spin (7), SU (4) and G 2 ⋉ R 8 for IIA backgrounds. We show that the spacetime of backgrounds with spinors that have stability subgroup K ⋉ R 8 is a pp-wave propagating in an eightdimensional manifold with a K-structure. The spacetime of backgrounds with K-invariant Killing spinors is a fibre bundle with fibre spanned by the orbits of two commuting null Killing vector fields and base space an eight-dimensional manifold which admits a K-structure. Type II T-duality interchanges the backgrounds with K-and K ⋉ R 8 -invariant Killing spinors. We show that the geometries of the base space of the fibre bundle and the corresponding space in which the pp-wave propagates are the same. The conformal symmetry of the world-sheet action of type II strings propagating in these N = 2 backgrounds can always be fixed in the light-cone gauge.

In this paper we develop tools to study families of non-selfadjoint operators L(ϕ), ϕ ∈ P , characterized by the property that the spectrum of L(ϕ) is (partially) simple. As a case study we consider the Zakharov-Shabat operators L(ϕ) appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials ϕ of Sobolev class H N , N ≥ 0, so that all small eigenvalues of L(ϕ) are simple, is path connected and dense.

We construct normalized differentials on families of curves of infinite genus. Such curves are used to investigate integrable PDE's such as the focusing Nonlinear Schrödinger equation.