We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new theory that have no classical analogues.

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G -manifold is a stratified symplectic space. Suppose $1\ra A\ra G\ra T\ra 1$ is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G -manifold with respect to A yields a Hamiltonian T -space. We show that if the A -moment map is proper, then the convexity theorem holds for such a Hamiltonian T -space, even when it is singular. We also prove that if, furthermore, the T -space has dimension 2dimT and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case.

For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a pair of mutual commutants) in the vertex algebra setting.

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classical equivariant cohomology, the latter being defined on the category of $G^*$ algebras a la H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call sg[t] algebras. In the geometrical setting, our principal example of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra structure but retains the structure of an sg[t] algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allows us to delineate a number of interesting geometric aspects of the G manifold M, sometimes in ways that are quite different from the classical theory.

This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we compute the coefficients of our rational function, and hence have all the necessary data to create a Schwarzian differential equation for each modular function. This equation can, in turn, be used to recover the modular function itself.