One – Loop Finiteness of the Four - Dimensional Donaldson - Nair - Schiff Non - Linear Sigma - Model 1

The most general four-dimensional non-linear sigma-model, having the second-order derivatives only and interacting with a background metric and an antisymmetric tensor field, is constructed. Despite its apparent non-renormalizability, just imposing the one-loop UV-finiteness conditions determines the unique model, which may be finite to all orders of the quantum perturbation theory. This model is known as the four-dimensional Donaldson-Nair-Schiff theory, which is a four-dimensional analogue of the standard two-dimensional Wess-Zumino-Novikov-Witten model, and whose unique finiteness properties and an infinite-dimensional current algebra have long been suspected. 1 Introduction. The two-dimensional Wess-Zumino-Novikov-Witten (WZNW) model [1] is the particular non-linear sigma-model (NLSM) whose target space is a group manifold, and the NLSM torsion to be represented by the WZ term paral-lelizes the group manifold. The WZNW model is a conformally invariant quantum field theory and, hence, it is finite to all orders of the quantum perturbation theory. It possesses on-shell the conserved affine currents which satisfy an infinite-dimensional affine algebra.