All (4,0): Sigma models with (4,0) off-shell supersymmetry

2006

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.

Journal of High Energy Physics, 2015

Briefly: using a novel (1, 1) superspace formulation of semichiral sigma models with 4D target space, we investigate if an extended supersymmetry in terms of semichirals is compatible with having a 4D target space with torsion. In more detail : semichiral sigma models have (2, 2) supersymmetry and Generalized Kähler target space geometry by construction. They can also support (4, 4) supersymmetry and Generalized Hyperkähler geometry, but when the target space is four dimensional indications are that the geometry is restricted to Hyperkähler. To investigate this further, we reduce the model to (1, 1) superspace and construct the extra (on-shell) supersymmetries there. We then find the conditions for a lift to (2, 2) super space and semichiral fields to exist. Those conditions are shown to hold for Hyperkähler geometries. The SU(2) ⊗ U(1) WZW model, which has (4, 4) supersymmetry and a semichiral description, is also investigated. The additional supersymmetries are found in (1, 1) superspace but shown not to be liftable to a (2, 2) semichiral formulation.

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.

2010

We study dualities in off-shell 4D N = 2 supersymmetric σ-models, using the projective superspace approach. These include (i) duality between the real O(2n) and polar multiplets; and (ii) polar-polar duality. We demonstrate that the dual of any superconformal σ-model is superconformal. Since N = 2 superconformal σmodels (for which target spaces are hyperkähler cones) formulated in terms of polar multiplets are naturally associated with Kähler cones (which are target spaces for N = 1 superconformal σ-models), polar-polar duality generates a transformation between different Kähler cones. In the non-superconformal case, we study implications of polar-polar duality for the σ-model formulation in terms of N = 1 chiral superfields. In particular, we find the relation between the original hyperkähler potential and its dual. As an application of polar-polar duality, we study self-dual models. 1 kuzenko@cyllene.uwa.edu.au 2 ulf.lindstrom@fysast.uu.se 3 unge@physics.muni.cz 8 Final comments 34 A Superconformal Killing vectors 35 B Tensor multiplet formulation for N = 2 σ-models with U(1)×U(1) symmetry 36 1 See [4, 5] for alternative approaches. General N = 2 supersymmetric σ-models in harmonic superspace and their dualities were studied in [6]

Journal of High Energy Physics, 2013

Semichiral sigma models with a four-dimensional target space do not support extended N = (4, 4) supersymmetries off-shell [1], . We contribute towards the understanding of the non-manifest on-shell transformations in (2, 2) superspace by analyzing the extended on-shell supersymmetry of such models and find that a rather general ansatz for the additional supersymmetry (not involving central charge transformations) leads to hyperkähler geometry. We give non-trivial examples of these models.

Journal of High Energy Physics, 2010

We study dualities in off-shell 4D N = 2 supersymmetric σ-models, using the projective superspace approach. These include (i) duality between the real O(2n) and polar multiplets; and (ii) polar-polar duality. We demonstrate that the dual of any superconformal σ-model is superconformal. Since N = 2 superconformal σmodels (for which target spaces are hyperkähler cones) formulated in terms of polar multiplets are naturally associated with Kähler cones (which are target spaces for N = 1 superconformal σ-models), polar-polar duality generates a transformation between different Kähler cones. In the non-superconformal case, we study implications of polar-polar duality for the σ-model formulation in terms of N = 1 chiral superfields. In particular, we find the relation between the original hyperkähler potential and its dual. As an application of polar-polar duality, we study self-dual models. 1 kuzenko@cyllene.uwa.edu.au 2 ulf.lindstrom@fysast.uu.se 3 unge@physics.muni.cz 8 Final comments 34 A Superconformal Killing vectors 35 B Tensor multiplet formulation for N = 2 σ-models with U(1)×U(1) symmetry 36 1 See [4, 5] for alternative approaches. General N = 2 supersymmetric σ-models in harmonic superspace and their dualities were studied in [6]

Proceedings of Proceedings of the Corfu Summer Institute 2014 — PoS(CORFU2014), 2015

2010

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.

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.