Ultraviolet Structure of Supersymmetric Non-Linear Sigma-Models

The applications of differential geometry to the general supersymmetric non-linear (sigma)-model on two space-time dimensions are considered. In particular three main topics are explored: (i) The construction of an adequate perturbation theory for these models. This is obtained using the background field expansion combined with the normal coordinate expansion. The necessity of these methods is most easily understood if one considers that the fields in general (sigma)-models take values on an arbitrary riemannian manifold and that the scattering matrix must be invariant under general reparameterizations of the manifold. Thus, one is led to the necessity of introducing a calculation scheme which can easily differentiate between the contributions to the S-matrix from those to the Green's functions. (ii) The characterization and classification of all possible Fermi-Bose symmetries which may appear in two space-time dimensions in the framework of non-linear (sigma)-models. It is proved that fermionic symmetries require necessarily the existence of complex structures on the manifold where the theory is defined. In particular, if the manifold is irreducible (i.e. it doesn't split locally into a product manifold), the only possible fermionic symmetries are supersymmetries; and that for this type of manifolds one can have at most four symmetries. The proof is based on the theory of complex manifolds and holonomy groups. (iii) As a consequence of the characterization theorem and the background field methods, one can give general criteria which determine what kind of counterterms will cancel the ultraviolet divergences of the S-matrix. If the model presents N = 4 supersymmetries, it is shown that all possible counterterms obey a linear partial differential equation (i.e. they are zero modes of the Lichnerowicz Laplacian). For models whose field manifold corresponds to self-dual gravitational instantaneous, it is possible to show that the equation mentioned above does not have solutions which might appear in perturbation theory; thus proving the ultraviolet finiteness of the S-matrix to all orders in perturbation theory. To my knowledge, this is the first time this type of results have been achieved.