Quantum cohomology of the Hilbert scheme of points in the plane

By A. Okounkov, R. Pandharipande. 2004

170 citations recorded by the source corpus.

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ2. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8550588f24cbda23bd778e4a9bb9d6191bdf7dda

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Lectures on K-theoretic computations in enumerative geometry (2015, background): “Such a relation indeed exists and has been already profitably used in cohomology [57,58,71,72].”
Elliptic stable envelopes (2016, result): “This extends, in particular, earlier results of [42, 51] on quantum cohomology of the Hilbert schemes of points of ADE surfaces.”
Localizing virtual cycles by cosections (2010, background): “The following Lemma was essentially proved in [18].”
The local Gromov-Witten theory of curves (2004, methodology): “where −q = e iu . The above formula was studied in [ 24 ] as the matrix of”
The Crepant Resolution Conjecture (2006, methodology): “Z2 ⊂ SU(2) in the next subsection, Z3 ⊂ SU(2) in [ 9 ], Z4 ⊂ SU(2) in [10],”
The local Donaldson–Thomas theory of curves (2005, methodology): “5 Proof of Proposition 24 We index the T –equivariant localization contributions to the invariants ̋ ŒI ˇ̌ ŒI ̨ n by graphs following [31].”
Supersymmetric vacua and Bethe ansatz (2009, unknown): “47 44 v2 [ he p- th ] 4 F eb 2 00 9 IHES-P/09/09 TCD-MATH-09-04 HMI-09-01 NSF-KITP-09-11 SUPERSYMMETRIC VACUA AND BETHE ANSATZ Nikita A. Nekrasova,1, and Samson L. Shatashvili1,2,3”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, background): “The argument of [21] uses • the proof of the MNOP conjecture for local curves established in the papers [5, 28, 29], • the proof of the MNOP conjecture for 3-folds An × P(1), where An is the holomorphic symplectic resolution of the standard An-surface singularity, in the papers [16, 19, 20].”
Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds (2008, background): “The triangle is established for A0 = C 2 in the series of papers [4, 29 , 30].”
Quantum Groups and Quantum Cohomology (2012, methodology): “iver with one vertex and one loop. In other words, we work out explicitly the quantum cohomology of the moduli spaces Mpr;nq of framed rank rtorsion free sheaves on C2, generalizing the previous work [103, 73] on the Hilbert schemes of points. 9 1 Introduction 10 1.1.2 Let X be a smooth quasi-projec”
Quantum Geometry and Quiver Gauge Theories (2013, methodology): “the quantum cohomology of Hilbert scheme of points on C(2) [186] in [168], and the relation to elliptic Macdonald operator was further studied in [187, 188].”
Curves on K3 surfaces and modular forms (2010, methodology): “y ﬁeld insertions has been established for toric V in [34]. The strategy of [34] is to match the capped Gromov-Witten and Donaldson-Thomas vertices. The 1-leg case was previously proven in [43] using [7, 42]. The Gromov-Witten and Donaldson-Thomas theory of An-resolutions was studied in [30, 32, 33]”