Ultraviolet Structure of Supersymmetric NonLinear SigmaModels
Abstract
The applications of differential geometry to the general supersymmetric nonlinear (sigma)model on two spacetime 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 Smatrix from those to the Green's functions. (ii) The characterization and classification of all possible FermiBose symmetries which may appear in two spacetime dimensions in the framework of nonlinear (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 Smatrix. 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 selfdual 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 Smatrix to all orders in perturbation theory. To my knowledge, this is the first time this type of results have been achieved.
 Publication:

Ph.D. Thesis
 Pub Date:
 1981
 Bibcode:
 1981PhDT........69A
 Keywords:

 Physics: Elementary Particles and High Energy