Geometry and beta functions for N = 2 matter models in two dimensions

Abstract

We study renormalizable nonlinear σ-models in two dimensions with N = 2 supersymmetry described in superspace in terms of chiral and complex linear superfields. The geometrical structure of the underlying manifold is investigated and the one-loop divergent contribution to the effective action is computed. The condition of vanishing β-function allows to identify a class of models which satisfy this requirement and possess N = 4 supersymmetry.

6D supersymmetric nonlinear sigma-models in 4D, Script N = 1 superspace

“…It is interesting to note that (4.28) is exactly the same constraint which was found in [19] from the condition of vanishing one-loop beta-function for a 2D CNM sigma-model with N = 4 supersymmetry. This implies that the resulting manifold is Ricci-flat and being four dimensional, it is necessarily hyper-Kähler [34,19].…”

“…Therefore, on-shell the CNM model describes a hyper-Kähler manifold as well. In particular, as also noted in [19], the duality Legendre transform acts on the manifold as a change of coordinates which is in general non-holomorphic (not preserving the complex structures).…”

“…This implies that the resulting manifold is Ricci-flat and being four dimensional, it is necessarily hyper-Kähler [34,19].…”

“…In the fourth chapter, we analyze the very difficult problem of deriving the geometry that arises in the case of directly using the CNM (chiral/non-minimal) [16]- [19] formulation without the starting point of projective superspace. The starting point for this mimics the techniques used in chapter two but includes now the complication to allow both chiral and complex linear superfields [20,21] (i.e.…”

Strongly correlated electron systems and quantum critical phenomena (11 April 2003, Troitsk, Moscow Region)

“…As we are going to show at the end of section 6 the condition (4.28) implies that the 4D target space geometry is hyper-Kähler. It is interesting to note that (4.28) is exactly the same constraint which was found in [19] from the condition of vanishing one-loop beta-function for a 2D CNM sigma-model with N = 4 supersymmetry. This implies that the resulting manifold is Ricci-flat and being four dimensional, it is necessarily hyper-Kähler [34,19].…”