We consider a four dimensional space-time symmetry which is a non trivial extension of the Poincaré algebra, different from supersymmetry and not contradicting a priori the well-known no-go theorems. We investigate some field theoretical aspects of this new symmetry and construct invariant actions for non-interacting fermion and noninteracting boson multiplets. In the case of the bosonic multiplet, where two-form fields appear naturally, we find that this symmetry is compatible with a local U (1) gauge symmetry, only when the latter is gauge fixed by a 't Hooft-Feynman term.

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We present a systematic approach to constructing current algebras based on non-semisimple groups. The Virasoro central charges corresponding to these current algebras are not, in general, given by integer numbers. The key point in this construction is that the bilinear form appearing in the current algebra can be different from the bilinear form used to raise and lower group indices. The action which realises this current algebra as its symmetry is also found. *

The problem of quantum equivalence between non-linear sigma models related by Abelian or non-Abelian T-duality is studied in perturbation theory. Using the anomalous Ward identity for Weyl symmetry we derive a relation between the Weyl anomaly coefficients of the original and dual theories. The analysis is not restricted to conformally invariant backgrounds. The formalism is applied to the study of two examples. The first is a model based on SU (2) non-Abelian T duality. The second represents a simple realization of Poisson-Lie T duality involving the Drinfeld double based on SU (2). In both cases quantum T duality is established at the 1-loop level.