Self-dual supersymmetric nonlinear sigma models

In four-dimensional N= 1 Minkowski superspace, general nonlinear σ-models with four-dimensional target spaces may be realised in term of CCL (chiral & complex linear) dynamical variables which consist of a chiral scalar, a complex linear scalar and their conjugate superfields. Here we introduce CCL σ-models that are invariant under U(1) "duality rotations" exchanging the dynamical variables and their equations of motion. The Lagrangians of such σ-models prove to obey a partial differential equation that is analogous to the self-duality equation obeyed by U(1) duality invariant models for nonlinear electrodynamics. These σ-models are self-dual under a Legendre transformation that simultaneously dualises (i) the chiral multiplet into a complex linear one; and (ii) the complex linear multiplet into a chiral one. Any CCL σ-model possesses a dual formulation given in terms of two chiral multiplets. The U(1) duality invariance of the CCL σ-model proves to be equivalent, in the dual chiral formulation, to a manifest U(1) invariance rotating the two chiral scalars. Since the target space has a holomorphic Killing vector, the σ-model possesses a third formulation realised in terms of a chiral multiplet and a tensor multiplet. The family of U(1) duality invariant CCL σ-models includes a subset of N = 2 supersymmetric theories. Their target spaces are hyper Kähler manifolds with a non-zero Killing vector field. In the case that the Killing vector field is triholomorphic, the σ-model admits a dual formulation in terms of a self-interacting off-shell N= 2 tensor multiplet. We also identify a subset of CCL σ-models which are in a one-to-one correspondence with the U(1) duality invariant models for nonlinear electrodynamics. The target space isometry group for these sigma models contains a subgroup U(1) × U(1). © 2013 SISSA, Trieste, Italy.