ar X iv : h ep - t h / 06 11 24 8 v 1 2 4 N ov 2 00 6 Generic N = 4 supersymmetric hyper - Kähler sigma models in D = 1

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. Introduction It has been known for a long time that the target space geometry of the sigma model is intimately related with the number of supersymmetries it possesses. In particular, Bruno Zumino showed that (2, 2) supersymmetry in D = 1 requires the bosonic part of a Lagrangian to be a Kähler manifold [1]. Later on, Alvarez-Gaume and Freedman [2] proved that (4, 4) supersymmetry further restricts the target space geometry to be hyper-Kähler (HK). Next, the analysis of the supersymmetric sigma-models with WessZumino terms [3] and heterotic (4, 0) supersymmetric sigma models [4, 5] brought about hyper-Kähler geometries with torsion (HKT). Apart from their evident application to non-linear sigma-models, HK and HKT geometries arise also in the moduli spaces for a certain class of black holes [5], in the target space of a bound state of a D-string and Dfive-branes, etc. Unfortunately, all these applications, though very interesting, are rather complicated. Moreover, the mathematical description of supersymmetric sigma models with HK and/or HKT target space geometries is quite involved. Therefore it seems to be a promising idea to simplify everything in such a way as to provide the simplest theory where the HK geometry arises as a consequence of supersymmetry, and where all main properties of the theory can understood. Clearly enough, supersymmetric mechanics should be a good choice, in this respect. The supersymmetric mechanics with N = 4 supersymmetry possesses a number of specific features which make it selected with respect, not only to its higher-dimensional counterparts, but also to mechanics with a different number of supersymmetries. Firstly, N = 4, D = 1 supersymmetry is rather simple. Moreover, just in the N = 4, D = 1 case the most general action may be easily written in terms of superfields as an integral over the whole superspace (in close analogy with (2, 2) supersymmetry in d = 2). Secondly, all known N = 4 supermultiplets in D = 1 are off-shell, so the corresponding actions can be written in standard superspace (see e.g. [7] and refs. therein). One should stress that just in N = 4, D = 1 superspace one may define a new class of nonlinear supermultiplets which contain a functional freedom in the defining relations [8, 9, 10]. Let us recall that we formulated the problem of how to describe N = 4 and N = 8, D = 1 sigma models with HK metrics in the target space in [11], advocating the use of nonlinear supermultiplets. Finally, in one dimension there is a nice duality between cyclic variables in Lagrangian and coupling constants. Indeed, if some one dimensional Lagrangian has a cyclic variable, say φ, then the corresponding conserved momentum pφ acquires a constant value m. Performing a Routh transformation over φ we will get a theory with a smaller number of bosonic fields but with a coupling constant m. Obviously, this procedure may be reversed to dualize the coupling constant m into a new bosonic field φ. Clearly enough, the resulting Lagrangian will possess an isometry with the Killing vector ∂/∂φ. In what follows we will heavily use just these features of supersymmetric mechanics. It is known that four dimensional bosonic hyper-Kähler manifolds with (at least) one isometry may be divided into two types that are in fact distinct from each other. The first kind, which is sometimes called translational (or triholomorphic), corresponds to a Killing vector with self-dual covariant derivatives. In the supersymmetric case the 1 The same type of bosonic target HKT geometry as in the D = 1 case was present in the N = 8, D = 1 analytic bi-harmonic superspace, see e.g. [6] and references therein. Here we closely follow [12].