A class of Osserman spaces

Journal of High Energy Physics, 2007

We review the projective-superspace construction of four-dimensional N = 2 supersymmetric sigma models on (co)tangent bundles of the classical Hermitian symmetric spaces.

Letters in Mathematical Physics, 2011

We discuss the conditions for extra supersymmetry in N = (2, 2) supersymmetric nonlinear sigma models described by one left and one right semi-chiral superfield and carrying a pair of non-commuting complex structures. Focus is on linear non-manifest transformations of these fields that have an algebra that closes off-shell. We solve the conditions for invariance of the action and show that a class of these solutions correspond to a bihermitian metric of signature (2, 2) and a pseudo-hyperkähler geometry of the target space. This is in contrast to the usual sector of bi-hermitian geometry with commuting complex structures where extra supersymmetries lead to bi-hypercomplex target space geometry.

Journal of High Energy Physics, 2012

Two-dimensional (2, 2) supersymmetric nonlinear sigma models can be described in (2, 2), (2, 1) or (1, 1) superspaces. Each description emphasizes different aspects of generalized Kähler geometry. We investigate the reduction from (2, 2) to (2, 1) superspace. This has some interesting nontrivial features arising from the elimination of nondynamical fields. We compare quantization in the different superspace formulations.

Commun Math Phys, 1987

We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a symplectic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

SSRN Electronic Journal

On 4-symmetric symplectic spaces, invariant almost complex structuresup to sign-arise in pairs. We exhibit some 4-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.

Communications in Mathematical Physics, 1987

We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a symplectic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

Russian Academy of Sciences Sbornik Mathematics, 2003

Let G/K be an irreducible Hermitian symmetric space of compact type with the standard homogeneous complex structure. Then the real symplectic manifold (T * (G/K), Ω) has the natural complex structure J −. We construct all Ginvariant Kähler structures (J, Ω) on homogeneous domains in T * (G/K) anticommuting with J −. Each such a hypercomplex structure, together with a suitable metric, defines a hyper-Kählerian structure. As an application, we obtain a new proof of the Harish-Chandra and Moore theorem for Hermitian symmetric spaces. Bibliography: 13 titles.

Communications in Mathematical Physics, 1988

We present new constructions of hyperkahler metrics along with the three new classes of N = 4 supermultiplets that they derive from. Further, we provide a general setting for understanding the constructions and give a detailed description of the multiplets in N = 2 and N = 4 superspace.

Journal of High Energy Physics, 2009

Quaternion Kähler manifolds are known to be the target spaces for matter hypermultiplets coupled to N = 2 supergravity. It is also known that there is a oneto-one correspondence between 4n-dimensional quaternion Kähler manifolds and those 4(n + 1)-dimensional hyperkähler spaces which are the target spaces for rigid superconformal hypermultiplets (such spaces are called hyperkähler cones). In this paper we present a projective-superspace construction to generate a hyperkähler cone M 4(n+1) H of dimension 4(n + 1) from a 2n-dimensional real analytic Kähler-Hodge manifold M 2n K . The latter emerges as a maximal Kähler submanifold of the 4n-dimensional quaternion Kähler space M 4n Q such that its Swann bundle coincides with M 4(n+1) H . Our approach should be useful for the explicit construction of new quaternion Kähler metrics. The results obtained are also of interest, e.g., in the context of supergravity reduction N = 2 → N = 1, or alternatively from the point of view of embedding N = 1 matter-coupled supergravity into an N = 2 theory.

Journal of High Energy Physics, 2004

We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N = 2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-Kähler geometry.