Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem. Furthermore, we recover a system that contains two types of the KP equation with self-consistent sources as special cases. Darboux and binary Darboux transformations are applied to generate solutions of the proposed hierarchies

Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider III(b) system from a quasi-Hamiltonian reduction of the internally fused double SU(n) x SU(n). In particular, the reduced spectral functions depending respectively on the first and second SU(n) factor of the double engender two toric moment maps on the III(b) phase space CP(n-1) that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z) mapping class group of the torus with one boundary component is shown to act on the quasi-Hamiltonian double by automorphisms and, upon reduction, the standard generator S of the mapping class group is proved to descend to the Ruijsenaars self-duality symplectomorphism that exchanges the toric moment maps. We give also two new presentations of this duality map: one as the composition of two Delzant symplectomorphisms and the other as the composition of three Dehn twist symplectomorphisms realized by Goldman twist flows. Through the well-known relation between quasi-Hamiltonian manifolds and moduli spaces, our results rigorously establish the validity of the interpretation [going back to Gorsky and Nekrasov] of the III(b) system in terms of flat SU(n) connections on the one-holed torus.Comment: Final version to appear in Nuclear Physics B, with simplified proof of Theorem 1, 56 page

AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of Kahler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.Comment: 56 pages; v2: minor changes, version to be published in Commun. Math. Phy

New compact forms of the trigonometric Ruijsenaars-Schneider system

The reduction of the quasi-Hamiltonian double of ${\mathrm{SU}}(n)$ that has been shown to underlie Ruijsenaars' compactified trigonometric $n$-body system is studied in its natural generality. The constraints contain a parameter $y$, restricted in previous works to $0<y < \pi/n$ because Ruijsenaars' original compactification relies on an equivalent condition. It is found that allowing generic $0<y<\pi/2$ results in the appearance of new self-dual compact forms, of two qualitatively different types depending on the value of $y$. The type (i) cases are similar to the standard case in that the reduced phase space comes equipped with globally smooth action and position variables, and turns out to be symplectomorphic to ${\mathbb{C}P^{n-1}}$ as a Hamiltonian toric manifold. In the type (ii) cases both the position variables and the action variables develop singularities on a nowhere dense subset. A full classification is derived for the parameter $y$ according to the type (i) versus type (ii) dichotomy. The simplest new type (i) systems, for which $\pi/n < y < \pi/(n-1)$, are described in some detail as an illustration.Comment: 31 page