Framed torsion-free sheaves on \(\mathbb {CP}^2\), Hilbert schemes, and representations of infinite dimensional Lie algebras
In the paper under review, the author presents geometric representations of Heisenberg and Clifford algebras on the equivariant cohomology of the moduli space of framed torsion free sheaves on the complex projective plane \(\mathbf{C}P^2\), and on the equivariant cohomology of the Hilbert scheme of points on the resolution \(X_r\) of the simple singularity \(\mathbf{C}^2/\text\textbf{Z}_r\). The main ideas are to exhibit explicit correspondences inside products of equivariant subvarieties of the relevant spaces, and to use equivariant localization to prove that the correspondences satisfy the defining relations of Heisenberg and Clifford algebras. \(\quad\)In Section 2 and 3, the author reviews basic facts about Heisenberg algebras, Clifford algebras and the boson-fermion correspondence. Quiver varieties were introduced in Sect.~4. Both the moduli space \(\mathcal M(r, n)\) of framed rank-\(r\) torsion free sheaves on \(\mathbf{C}P^2\) and the Hilbert scheme \((X_r)^{[n]}\) of points on the resolution \(X_r\) are special cases of quiver varieties. Torus actions on \(\mathcal M(r, n)\) and \((X_r)^{[n]}\) are defined in Section 5; the fixed loci are also identified. In Section 6, the author studies the equivariant cohomology of \(\mathcal M(r, n)\) and \((X_r)^{[n]}\), and localization of certain correspondences. Section 7 and 8 are devoted to geometric constructions of the Clifford algebras and the Heisenberg algebras respectively. In Section 9, it is proved that geometric constructions satisfy the defining relations of Heisenberg and Clifford algebras. In Section 10, the author describes a geometric interpretation of the boson-fermion correspondence. In Section 11, relations between the moduli spaces \(\mathcal M(r, n)\) and \((X_r)^{[n]}\) are investigated.
scientific article; zbMATH DE number 5835525
scientific article