Global sections of the chiral de Rham complex for Calabi-Yau and hyperkahler manifolds

Abstract: For any complex manifold M, the chiral de Rham complex is a sheaf of vertex algebras on M that was introduced in 1998 by Malikov, Schechtman, and Vaintrob. It is N-graded by conformal weight, and the weight zero piece coincides with the ordinary de Rham sheaf. When M is a Calabi-Yau manifold with holonomy group SU(d), it was shown by Ekstrand, Heluani, Kallen and Zabzine that the algebra of global sections \Omega^{ch}(M) contains a certain vertex algebra defined by Odake which is an extension of the N=2 superconformal algebra. When M is a hyperkahler manifold, it was shown by Ben-Zvi, Heluani, and Szczesny that \Omega^{ch}(M) contains the small N=4 superconformal algebra. In this talk, we will show that in both cases, these subalgebras are actually the full algebras of global sections. In an earlier work, Bailin Song has shown that the global section algebra can be identified with a certain subalgebra of a free field algebra which is invariant under the action of an infinite-dimensional Lie algebra of Cartan type. The key observation is that this algebra can be described using the arc space analogue of Weyl's first and second fundamental theorems of invariant theory for the special linear and symplectic groups.

This is a joint work with Bailin Song.

algebraic geometrydifferential geometryquantum algebrasymplectic geometry

Audience: researchers in the topic

Boston University Geometry/Physics Seminar

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