Quantum cohomology of the Hilbert scheme of points on 𝒜_{𝓃}-resolutions

Maulik, Davesh; Oblomkov, Alexei

doi:10.1090/S0894-0347-09-00632-8

Quantum cohomology of the Hilbert scheme of points on $\mathcal{A}_n$ -resolutions

Authors: Davesh Maulik and Alexei Oblomkov
Journal: J. Amer. Math. Soc. 22 (2009), 1055-1091
MSC (2000): Primary 14N35
DOI: https://doi.org/10.1090/S0894-0347-09-00632-8
Published electronically: March 24, 2009
MathSciNet review: 2525779
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Abstract: We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type $A_{n}$ singularities. The operators encoding these invariants are expressed in terms of the action of the the affine Lie algebra $\widehat{\mathfrak{gl}}(n+1)$ on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of $A_{n}\times\mathbf{P}^1$ . We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of types and .

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