The Geometry of Supersymmetric D = 2 Nonlinear Sigma Models

This paper will consider supersymmetric nonlinear sigma models in two space-time dimensions. First, the rich variety of possible supersymmetries in two dimensions will be discussed. Next, a slight detour into some complex geometry will be taken to gather some useful definitions and results. Finally, it will be shown that requiring the sigma model to have extended supersymmetry imposes various geometric constraints on the target manifold of the model. Nonlinear sigma models are the quantum field theories of harmonic maps from space-time into a Riemannian manifold M. In d = 2, a scalar field is dimensionless so a Lagrangian of the form L = ∫ dxgij(φ)∂μφ∂φ is renormalizable. As any function, gij(φ), leads to a renormalizable theory, this class of field theories contains an infinite number of marginal parameters. To restrict the possible parameters, symmetries are imposed. This is generally accomplished by restricting the fields to take values in a Riemannian manifold. Classic examples are S and CP. Supersymmetric nonlinear sigma models enlarge the set of symmetries of the theory to include supersymmetries. This can be done by writing the above Lagrangian in terms of chiral superfields, for instance. The Lagrangian for the scalar components of the superfields will then be the same as above. One reason to study these models is their application to string theory. One can view the d = 2 space-time as a string world-sheet. Extended world-sheet supersymmetries have had two different uses in string theory. The heterotic string has only a gauged (1, 1) or (1, 0) world-sheet supersymmetry, but the corresponding (1, 1) or (1, 0) sigma model on a suitable background can have extra rigid world-sheet supersymmetries. This fact has played a central role in the study of compactifications of the theory. Sigma models have also played a central role in the study of N = 2 strings, or more generally, strings with (2, 0), (2, 1), or (2, 2) world-sheet supersymmetry. For instance, the heterotic sigma models which describe the target spaces of (2, 1) strings require a particular geometry for the target space. It is Hermitean with torsion, and the field equations imply that the curvature with torsion is self-dual in four dimensions, or satisfies generalised self-duality equations in higher dimensions. For a more thorough discussion of the applications of sigma models to string theory, see [1] and references therein. The main point is that the extended supersymmetries impose restrictions on the geometry of the target manifold. This, in turn, allows one to make stronger statements about the quantum theory, with regards to vacuum structure and ultra-violet behavior in particular.