Complex structures, duality and WZW-models in extended superspace

Abstract

We find the complex structure on the dual of a complex target space. For N = (2, 2) systems, we prove that the space orthogonal to the kernel of the commutator of the left and right complex structures is always integrable, and hence the kernel is parametrized by chiral and twisted chiral superfield coordinates. We then analyze the particular case of SU (2) × SU (2), and are led to a new N = 2 superspace formulation of the SU (2) × U (1) WZW-model.

The biHermitian topological sigma model

“…e) The complex structures K ± a b are parallel with respect to the connections ∇ ±a 5) where the connection coefficients Γ ± a bc are given by…”

“…In a classic paper, Gates, Hull and Roček [3] showed that, for a 2-dimensional sigma model, the most general target space geometry allowing for (2, 2) supersymmetry was biHermitian or Kaehler with torsion geometry. This is characterized by a Riemannian metric g ab , two generally non commuting complex structures K ± a b and a closed 3-form H abc , such that g ab is Hermitian with respect to both the K ± a b and the K ± a b are parallel with respect to two different metric connections with torsion proportional to ±H abc [4][5][6][7]. This geometry is more general than that considered by Witten, which corresponds to the case where K + a b = ±K − a b and H abc = 0.…”

Off-shell formulation of N = 2 non-linear σ-models

“…An important result obtained in [9] states that ker(J −J ) and ker(J +J) are always integrable. This allows us to parametrize these kernels by chiral and twisted-chiral fields resp.…”

“…In [9] an implicit description of the SU(2) × U(1) WZW model was given but now in terms of a semi-chiral multiplet. Here we give the explicit description of the model using one semi-chiral multiplet.…”

Supersymmetric Sigma Model Geometry

“…See [65] for a more detailed discussion of the above coordinatization. Furthermore, as discussed for GKG in the previous section, the commuting complex structures imply the existence of a local product structure…”