Spectra of Conformal Sigma Models

In this thesis the spectra of conformal sigma models defined on (generalized) symmetric spaces are analysed. The spaces where sigma models are conformal without the addition of a Wess-Zumino term are supermanifolds, in other words spaces that include fermionic directions. After a brief review of the general construction of vertex operators and the background field expansion, we compute the diagonal terms of the one-loop anomalous dimensions of sigma models on semi-symmetric spaces. We find that the results are formally identical to the symmetric case. However, unlike for sigma models on symmetric spaces, off diagonal terms that lead to operator mixing are also present. These are not computed here. We then present a detailed analysis of the one-loop spectrum of the supersphere S^3|2 sigma model as one of the simplest examples. The analysis illustrates the power and simplicity of the construction. We use this data to revisit a duality with the OSP(4|2) Gross-Neveu model that was proposed by Candu and Saleur. With the help of a recent all-loop result for the anomalous dimension of 1/2BPS operators of Gross-Neveu models, we are able to recover the entire zero-mode spectrum of the supersphere model. We also argue that the sigma model constraints and its equations of motion are implemented correctly in the Gross-Neveu model, including the one-loop data. The duality is further supported by a new all-loop result for the anomalous dimension of the ground states of the sigma model. However, higher-gradient operators cannot be completely recovered. It is possible that this discrepancy is related to a known instability of the sigma model. The instability of sigma models is due to symmetry preserving high-gradient operators that become relevant at arbitrarily small values of the coupling. This feature has been observed long ago in one-loop calculations of the O(N)-vector model and soon been realized to be a generic property of sigma models that persists to higher loop orders. A similar instability has been observed for Gross-Neveu models which can be seen as a certain deformation of WZNW models at level k = 1. Recently, Ryu et al. suggested that the psl(N|N) Gross-Neveu models might be free of relevant high-gradient operators. They tested this proposal at one-loop level for a certain class of invariant operators. We extend the result to all invariant operators and all loops for the psl(2|2) Gross-Neveu model. Additionally, we determine the spectrum of the BPS sector at infinite coupling and observe that all scaling weights become half-integer. Evidence for a proposed duality with the CP^1|2 sigma model is not found. We conclude with a brief discussion of marginal deformations of Kazama-Suzuki models