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

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

Journal of High Energy Physics, 2005

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.

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.

A brief review of supersymmetric non-linear sigma models and generalized complex geometry

2006

This is a review of the relation between supersymmetric non-linear sigma models and target space geometry. In particular, we report on the derivation of generalized K\"ahler geometry from sigma models with additional spinorial superfields. Some of the results reviewed are: Generalized complex geometry from sigma models in the Lagrangian formulation; Coordinatization of generalized K\"ahler geometry in terms of chiral, twisted chiral and semi-chiral superfields; Generalized K\"ahler geometry from sigma models in the Hamiltonian formulation.

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.

Pseudo-Hyperkähler Geometry and Generalized Kähler Geometry

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.

Generalized Kähler geometry in (2, 1) superspace

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.

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.

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"

Sigma models with off-shell N = (4, 4) supersymmetry and noncommuting complex structures

Journal of High Energy Physics, 2010

We describe the conditions for extra supersymmetry in N = (2, 2) supersymmetric nonlinear sigma models written in terms of semichiral superfields. We find that some of these models have additional off-shell supersymmetry. The (4, 4) supersymmetry introduces geometrical structures on the target-space which are conveniently described in terms of Yano f -structures and Magri-Morosi concomitants. On-shell, we relate the new structures to the known bi-hypercomplex structures.

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.