Quantum cohomology of the Hilbert scheme of points on A_n-resolutions

By Davesh Maulik, Alexei Oblomkov. 2008

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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 affine Lie algebra \hat{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 x P^1. We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type D and E.

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

View Paper: https://www.semanticscholar.org/paper/b198f29fcf9b21be8687a7d7197419d20a8aadeb

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Curve Counting on K3 × E, The Igusa Cusp Form χ10, and Descendent Integration (2016, methodology): “. The A1 resolution. Let E Pairs B = [q −1]E and EHilb B = [q −1]EHilb be the restriction of EHilb and EPairs to the case of the class β 0 = B.14 The corresponding local case was considered before in [22, 23]. Deﬁne operators E(r) B by 1S |E (r) B 1S = y (1+y)2 δ0r [pm(γ),E (r) B ] = γ,B (−y)−m/2 −(”