Laumon Spaces and the Calogero-Sutherland Integrable System
Andrei Negut
TL;DR
The paper proves Braverman's conjecture that the generating function $Z(m)$, built from equivariant Chern polynomials of tangent bundles on Laumon quasiflag spaces, is (up to an explicit factor) an eigenfunction of the quantum trigonometric Calogero-Sutherland Hamiltonian. It achieves this by constructing a geometric operator $A(m)$ from a vector bundle $E$ on the Laumon moduli space product, relating its action to $\mathfrak{sl}_n$ intertwiners and generalized characters, and deriving an explicit expression for the generating function in terms of the CS eigenfunction $Y_{a/x,m}$ and Weyl denominator factors. The main result yields $Z(m)=Y_{a/x,m}\,e^{-a/x}\,\big(\prod_{\alpha\in R^+}(1-e^{-{\alpha}})\big)^{-(m+1)}$, with a finite-dimensional Toda limit as $m\to\infty$, establishing a precise link between Laumon geometry, integrable systems, and representation theory. The techniques also illuminate connections to equivariant quantum cohomology and gauge theory, and confirm the finite-m correspondence conjectured by Braverman in the ${\mathfrak{sl}}_n$ setting.
Abstract
This paper contains a proof of a conjecture of Braverman concerning Laumon quasiflag spaces. We consider the generating function Z(m), whose coefficients are the integrals of the equivariant Chern polynomial (with variable m) of the tangent bundles of the Laumon spaces. We prove Braverman's conjecture, which states that Z(m) coincides with the eigenfunction of the Calogero-Sutherland hamiltonian, up to a simple factor which we specify. This conjecture was inspired by the work of Nekrasov in the affine \hat{sl}_n setting, where a similar conjecture is still open.