Yangians and cohomology rings of Laumon spaces

Abstract

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of G L n . We construct the action of the Yangian of 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 sl n [s ±1 , 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 M n,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is 574 B. Feigin et al.naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of gl n naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on M n,d is the image of a noncommutative power sum in Z .

The stringy instanton partition function

“…This introduces surface operators in the gauge theory [62] and provides a set-up to compute the quantum cohomology of their moduli spaces, such as for example Laumon spaces [65] and partial flag varieties [66]. Our approach should be compared with the results of [67].…”

Affine sl(N)conformal blocks from $ \mathcal{N} = 2 $ SU(N) gauge theories

“…It was further realised in [1] that the technology to compute such instanton partition functions already exists in the mathematics literature [23][24][25][26]. Using these results several checks of the proposed relation were performed.…”

“…In a recent paper [1] Alday and Tachikawa proposed that the formalism needed to determine the instanton partition function in the presence of a full surface operator in an SU(N ) theory has already been developed in the mathematical literature [23][24][25][26]. (Strictly speaking, it is not completely obvious that the problem solved by the mathematicians is really equivalent to the physics problem, but this is believed to be the case.…”

“…Similarly, the Coulomb moduli are assumed to have the same property: a i ≡ a i+N . The character for a bifundamental multiplet can then be written [1,26] …”

“…For a gauge group with a single SU(N ) factor the result is of the form 9) where the instanton numbers k i are given by [1,26] …”