The background field method and the ultraviolet structure of the supersymmetric nonlinear σ-model

N=2σ -model action on non(anti)commutative superspace

“…Supersymmetric extension of bosonic sigma models unfolded new avenues in the study of complex geometry. In particular, extension to N = 2 supersymmetry forces the target manifold to be Kähler [1]- [4], which plays a crucial role in showing the consistency of these sigma models at the quantum level. The importance of the sigma model approach to the study of string compactifications on Ricci-flat Kähler manifolds (otherwise Calabi-yau manifolds) and also string propagation in arbitrary background fields is well known.…”

Gravitational dressing of N = 2 σ-models beyond leading order

“…The model with N = 1 supersymmetry is for several aspects very similar to the bosonic one. On a flat two-dimensional worldsheet it has been studied at the perturbative level using the quantum-background field expansion in normal coordinates [6] and the β-function has been computed up to high orders in the loop expansion. The leading contribution, proportional to the Ricci tensor, is at one loop, while the next to the leading nonvanishing correction is at four loops [7].…”

Bosonic string in affine-metric curved space

“…In the bosonic case a metric and a connection structures are used. Riemannian manifolds are considered as a field manifold for usual nonlinear sigma-model [1,2]. The connection structure of this manifold is uniquely constructed from metric, i.e.…”

“…Therefore it is surprising that the counterterms of the sigma model with affine-metric manifold differ from counter terms of the sigma model with Riemannian manifold [6]. This difference can not be reduced to the metric redefenition caused by infinitesimal coordinate transformation [2] or to the nonlinear renormalization of the quantum fields [8]. In the paper [6], the counter terms are calculated for conventional sigma model without assuming a metric connection for the geodesic line equation in covariant background field method.…”