Sigma models with a Wess-Zumino term in twistor spaces

Idrisse Khemar (1)

1 IECN - Institut Élie Cartan de Nancy

We characterize the Riemannian manifolds whose the twistor space satisfies the geometric properties necessary to the existence of some sigma model with a Wess-Zumino term on this twistor space. We prove that these manifolds are space forms. Then we study the Riemannian manifolds for which there exists a subbundle of the twistor space which satisfies these geometric properties and prove that in most cases these manifolds are locally homogeneous. In our study, we are led to prove some theorems about metric connections with parallel curvature: we prove for example that a metric connection with parallel curvature and with restricted holonomy group $SO(n)$ must be the Levi-Civita connection and therefore the Riemannian manifold is a space form. We also propose a general method to study metric connections with parallel curvature.

Mots clés: Sigma model; Variational formulation; twistor space; metric connection; skew-symmetric torsion; Geometric non linear PDE; Variational formulation.

Domaines: Géométrie différentielle [math.DG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Idrisse Khemar. Sigma models with a Wess-Zumino term in twistor spaces. 2012. ⟨hal-00767001⟩