Abstract
We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of $n$ points of $\mathbb{C}^{2}$ . Since the equivariant quantum cohomology, computed by Okounkov and Pandharipande [Invent. Math. 179 (2010), 523–557], is semisimple, the higher genus theory is determined by an $\mathsf{R}$ -matrix via the Givental–Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required $\mathsf{R}$ -matrix by explicit data in degree $0$ . As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme $\mathsf{Hilb}^{n}(\mathbb{C}^{2})$ and the Gromov–Witten/Donaldson–Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined by Okounkov and Pandharipande [Transform. Groups 15 (2010), 965–982]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov–Witten theory of the symmetric product $\mathsf{Sym}^{n}(\mathbb{C}^{2})$ is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].
Highlights
The R-matrix of the Cohomological Field Theories (CohFTs) associated to the local Donaldson–Thomas theories of curves is proven to coincide with the R-matrix of the T-equivariant Gromov–Witten theory of Hilbn(C2) determined in Theorem 1
An R-matrix associated to the formal Frobenius manifold (V, η) is a matrix series of the form (2.4) which determines a solution (2.3) of the quantum differential equation (2.2) and satisfies the symplectic condition (2.5)
We write {uμ ∈ Acl[[V∗]]}μ∈Part(n) for the unique canonical coordinates of (V, η) satisfying that uμ(0) = 0 and ∂/∂uμ is an idempotent. (Acl denotes the algebraic closure of the field of fractions of A.) By [11, Proposition 1.1], the quantum differential equation associated to the formal Frobenius manifold (V, η)
Summary
The R-matrix of the T-equivariant Gromov–Witten theory of Hilbn(C2) is uniquely determined from the T-equivariant genus 0 theory by the divisor equation and the degree 0 invariants μ. Theorem 1 can be equivalently stated in the following form: the R-matrix of the T-equivariant Gromov–Witten theory of Hilbn(C2). In genus 0, the equivalence of the T-equivariant Gromov–Witten theories of Hilbn(C2) and the orbifold Symn(C2) was proven in [1]. The R-matrix of the CohFT associated to the local Donaldson–Thomas theories of curves is proven to coincide with the R-matrix of the T-equivariant Gromov–Witten theory of Hilbn(C2) determined in Theorem 1. The R-matrices of the CohFTs associated to Symn(C2) and the local Gromov–Witten theories of curves are straightforward to match (with determination by Theorem 3). We do not explore here the interesting geometry of the (a, b)-level structure over the moduli space of genus g curves.)
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