Two-dimensional twisted sigma models and chiral differential operators
We explore a variety of two-dimensional twisted sigma models at the perturbative level in an attempt to furnish purely physical interpretations of various mathematical theories involving a??Chiral Differential Operatorsa?? (or CDOa??s) defined and constructed by mathematicians in recent times. In this thesis, we consider the following four cases.Firstly, we study a twisted version of the two-dimensional (0; 2) heterotic sigma model on a holomorphic gauge bundle E over a complex, hermitian manifold X. We show that the model can be naturally described in terms of the mathematical theory of CDOa??s.Secondly, we study the twisted heterotic sigma model at the (2; 2) locus. We show that the resulting sigma model is a half-twisted variant of Wittena??s topological A-model, which can be given a purely mathematical description in terms of the sheaf of a certain CDO called the Chiral de Rham Complex (or CDR).Thirdly, we study the half-twisted sigma model on a complex orbifold X/G, where G is an isometry group of X. Via this orbifold sigma model, we obtain a purely physical interpretation of a recently constructed mathematical theory of CDR on orbifolds.Finally, we study the half-twisted sigma model coupled to a non-dynamical gauge field with Kahler target space X being a smooth G-manifold. In doing so, we arrive at a purely physical interpretation of the equivariant cohomology of the CDR, recently defined by mathematicians, called the a??chiral equivariant cohomologya??.Via the math-physics connection unveiled, we find that various physical features of the above sigma models can be described in terms of interesting and novel mathematical ideas. Conversely, several results in the mathematical literature now lend themselves to simple physical explanations. The work in this thesis therefore opens up new and exciting possibilities for both mathematics and physics.
twisted sigma models, chiral differential operators
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