Vertex Operators, Grassmannians, and Hilbert Schemes

Details Vertex Operators, Grassmannians, and Hilbert Schemes Vertex Operators, Grassmannians, and Hilbert Schemes

2009-10-29

arxiv.org/abs/0910.5528v3

Mathematics

Representation Theory

20 pages, 0 figures

Scientific paper

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.

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