B.R.S. renormalisation of some on-shell closed algebras of symmetry transformations : N=2 and 4 supersymmetric non-linear sigma models

TL;DR

This paper investigates the renormalisability of N=2 and N=4 supersymmetric non-linear sigma models on Kähler spaces, revealing potential anomalies in N=2 models and confirming all-orders renormalisability for N=4 models.

Contribution

It provides a cohomological B.R.S. analysis showing potential anomalies in N=2 models and rigorously proves all-orders renormalisability of N=4 models.

Findings

Potential anomaly candidate in N=2 models on compact Ricci-flat Kähler spaces.

Anomaly candidate disappears in compact homogeneous Kähler cases.

Rigorous proof of all-orders renormalisability for N=4 supersymmetric sigma models.

Abstract

We analyse with the algebraic, regularisation independant, cohomological B.R.S. methods, the renormalisability of torsionless N=2 and N= 4 supersymmetric non-linear $\si$ models built on K\"ahler spaces. Surprisingly enough with respect to the common wisdom, in the case of N=2 supersymmetry, we obtain an anomaly candidate, at least in the compact K\"ahler Ricci-flat case. If its coefficient does differ from zero, such anomaly would imply the breaking of global N=2 supersymmetry and get into trouble some schemes of superstring compactification as such non-linear $\si$ models offer candidates for the superstring vacuum state. In the compact homogeneous K\"ahler case, as expected, the anomaly candidate disappears. The same phenomena occurs when one enforces N=4 supersymmetry : in that case, we obtain the first rigorous proof of the expected all-orders renormalisability -`` in the space of metrics"- of the corresponding non-linear $\si$ models. PAR/LPTHE/94-11