Yangians and cohomology rings of Laumon spaces

By Boris Feigin, Michael Finkelberg, Andrei Neguț, L. Rybnikov. 2008

87 citations recorded by the source corpus.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of $${\mathfrak{sl}_n}$$ in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of $${\mathfrak{sl}_n[s^{\pm1},t]}$$) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space $${\mathfrak{M}_{n,d}}$$ of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of $${\mathfrak{gl}_n}$$ naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on $${\mathfrak{M}_{n,d}}$$ is the image of a noncommutative power sum in Z.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/416cf079ee1a080494d204e65ccd0d4250ce51f3

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Citations in the corpus

Equivariant K-theory of Hilbert schemes via shuffle algebra (2009, background): “This work was motivated by [3] and [11].”
Instanton counting with a surface operator and the chain-saw quiver (2011, methodology): “In this case, the corresponding objects in algebraic geometry is a rank-N torsion-free sheaves on CP(1)×CP(1) with coordinates (z, w), with framing given at {z = ∞} ∪ {w = ∞} and with parabolic structure of type P given at {w = 0}, whose precise definition can be found in [29, 30].”
Affine SL(2) Conformal Blocks from 4d Gauge Theories (2010, background): “6 Strictly speaking, it is only proved that there is a natural action of affine SL (N ) on H S .T his decomposition is conjectured in [ 15 ].”