Generalized Kähler geometry in (2, 1) superspace (original) (raw)

Generalized Kahler geometry and manifest N=(2,2) supersymmetric nonlinear sigma-models

Journal of High Energy Physics

Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N = (2, 2) nonlinear sigma-models. The most direct relation is obtained at the N = (1, 1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N = (2, 2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike the corresponding ordinary complex structures) and describe a Generalized Kähler geometry. The metric, B-field and generalized complex structures are all determined in terms of a potential K.

Generalized Kähler geometry and manifest Script N = (2,2) supersymmetric nonlinear sigma-models

Journal of High Energy Physics, 2005

The geometry of supersymmetric sigma-models

We review non-linear σ-models with (2,1) and (2,2) supersymmetry. We focus on off-shell closure of the supersymmetry algebra and give a complete list of (2, 2) superfields. We provide evidence to support the conjecture that all N = (2, 2) non-linear σ-models can be described by these fields. This in its turn leads to interesting consequences about the geometry of the target manifolds. One immediate corollary of this conjecture is the existence of a potential for hyper-Kähler manifolds, different from the Kähler potential, which does not only allow for the computation of the metric, but of the three fundamental twoforms as well. Several examples are provided: WZW models on SU (2) × U (1) and SU (2) × SU (2) and four-dimensional special hyper-Kähler manifolds.

Generalized Kähler Geometry from Supersymmetric Sigma Models

Letters in Mathematical Physics, 2006

We give a physical derivation of generalized Kähler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri [10] regarding the equivalence between generalized Kähler geometry and the bi-hermitean geometry of Gates-Hull-Roček . When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.

Gauged (2,2) sigma models and generalized Kähler geometry

Journal of High Energy Physics, 2007

We gauge the (2, 2) supersymmetric non-linear sigma model whose target space has bihermitian structure (g, B, J ±) with noncommuting complex structures. The bihermitian geometry is realized by a sigma model which is written in terms of (2, 2) semichiral superfields. We discuss the moment map, from the perspective of the gauged sigma model action and from the integrability condition for a Hamiltonian vector field. We show that for a concrete example, the SU(2) × U(1) WZNW model, as well as for the sigma models with almost product structure, the moment map can be used together with the corresponding Killing vector to form an element of T ⊕ T * which lies in the eigenbundle of the generalized almost complex structure. Lastly, we discuss T-duality at the level of a (2, 2) sigma model involving semi-chiral superfields and present an explicit example.

Supersymmetric Sigma Model Geometry

Symmetry, 2012

This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces.

Generic supersymmetric hyper-Kähler sigma models in

Physics Letters B, 2007

We analyse the geometry of four-dimensional bosonic manifolds arising within the context of N = 4, D = 1 supersymmetry. We demonstrate that both cases of general hyper-Kähler manifolds, i.e. those with translation or rotational isometries, may be supersymmetrized in the same way. We start from a generic N=4 supersymmetric three-dimensional action and perform dualization of the coupling constant, initially present in the action. As a result, we end up with explicit component actions for N = 4, D = 1 nonlinear sigma-models with hyper-Kähler geometry (with both types of isometries) in the target space. In the case of hyper-Kähler geometry with translational isometry we find that the action possesses an additional hidden N = 4 supersymmetry, and therefore it is N = 8 supersymmetric one.

Generalized Kähler Manifolds and Off-shell Supersymmetry

Communications in Mathematical Physics, 2006

We solve the long standing problem of finding an off-shell supersymmetric formulation for a general N = (2, 2) nonlinear two dimensional sigma model. Geometrically the problem is equivalent to proving the existence of special coordinates; these correspond to particular superfields that allow for a superspace description. We construct and explain the geometric significance of the generalized Kähler potential for any generalized Kähler manifold; this potential is the superspace Lagrangian.

Homogeneous K�hler manifolds: Paving the way towards new supersymmetric sigma models

Communications in Mathematical Physics, 1986

Homogeneous Kahler manifolds give rise to a broad class of supersymmetric sigma models containing, as a rather special subclass, the more familiar supersymmetric sigma models based on Hermitian symmetric spaces. In this article, all homogeneous Kahler manifolds with semisimple symmetry group G are constructed, and are classified in terms of Dynkin diagrams. Explicit expressions for the complex structure and the Kahler structure are given in terms of the Lie algebra cj of G. It is shown that for compact G, one can always find an Einstein-Kahler structure, which is unique up to a constant multiple and for which the Kahler potential takes a simple form. * On leave of absence from Fakultat fur Physik der Universitat Freiburg, FRG 1 The term "homogeneous space" is synonymous for "coset space," and similarly, the term "Hermitian symmetric space" is synonymous for "symmetric Kahler manifold"

New extended superconformal sigma models and quaternion Kähler manifolds

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.