[Paper Review] Orbifold quantum cohomology of the symmetric product of A_r

Wan Keng Cheong|ArXiv.org|Oct 5, 2009

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper computes the equivariant orbifold quantum cohomology of the symmetric product stack $[\mathrm{Sym}^n(\mathcal{A}_r)]$, showing that divisor operators are determined by one-part double Hurwitz numbers. Under the nonderogatory conjecture, it establishes a full isomorphism between the quantum cohomology of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ and $\mathrm{Hilb}^n(\mathcal{A}_r)$, confirming the Crepant Resolution Conjecture and completing a tetrahedron of equivalences with relative Gromov-Witten theory of $\mathcal{A}_r \times \mathbb{P}^1$. The key result is an explicit, closed-form expression for 3-point extended Gromov-Witten invariants via combinatorial Hurwitz numbers.

ABSTRACT

Let A_r be the minimal resolution of the cyclic quotient singularity C^2/Z_{r+1}. We study the equivariant quantum cohomology ring of the n-fold symmetric product stack [Sym^n(A_r)] of A_r. We calculate the operators of quantum multiplication by divisor classes. Under the assumption of the nonderogatory conjecture, these operators completely determine the ring structure, which provides an affirmative answer to the Crepant Resolution Conjecture on [Sym^n(A_r)] and Hilb^n(A_r). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Sym^n(A_r)]/Hilb^n(A_r) and the relative Gromov-Witten/Donaldson-Thomas theories of A_r x P^1.

Motivation & Objective

• To compute the equivariant orbifold quantum cohomology ring of $[\mathrm{Sym}^n(\mathcal{A}_r)]$, the symmetric product stack of the minimal resolution of an $A_r$ surface singularity.
• To establish a precise isomorphism between the quantum cohomology of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ and $\mathrm{Hilb}^n(\mathcal{A}_r)$, the Hilbert scheme of $n$ points on $\mathcal{A}_r$, under the nonderogatory conjecture.
• To complete a tetrahedron of equivalences linking the Gromov-Witten theories of $[\mathrm{Sym}^n(\mathcal{A}_r)]$, $\mathrm{Hilb}^n(\mathcal{A}_r)$, and the relative Gromov-Witten/Donaldson-Thomas theories of $\mathcal{A}_r \times \mathbb{P}^1$.

Proposed method

• Uses localization techniques on the moduli space of twisted stable maps to compute 2-point extended Gromov-Witten invariants of $[\mathrm{Sym}^n(\mathcal{A}_r)]$.
• Reduces the computation of quantum multiplication by divisor classes to counting branched covers of rational curves via virtual localization and fixed point contributions.
• Employs one-part double Hurwitz numbers—explicitly computed via Goulden-Jackson-Vakil formulas—as the key combinatorial invariant encoding the quantum product structure.
• Applies the nonderogatory conjecture to ensure that divisor operators fully determine the orbifold quantum product.
• Constructs a linear isomorphism $L$ between Chen-Ruan cohomology of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ and cohomology of $\mathrm{Hilb}^n(\mathcal{A}_r)$, preserving gradings, Poincaré pairings, and quantum product by divisors.
• Uses recursive WDVV-type relations on extended Gromov-Witten invariants to prove that the isomorphism $L$ preserves all multi-point functions, thereby confirming the Crepant Resolution Conjecture.

Experimental results

Research questions

• RQ1 Can the orbifold quantum cohomology of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ be fully computed using combinatorial invariants?
• RQ2 Does the Crepant Resolution Conjecture hold for $[\mathrm{Sym}^n(\mathcal{A}_r)]$ and $\mathrm{Hilb}^n(\mathcal{A}_r)$, with the quantum product structure preserved under a linear isomorphism?
• RQ3 Is there a complete equivalence between the Gromov-Witten theory of $[\mathrm{Sym}^n(\mathcal{A}_r)]$, the Hilbert scheme $\mathrm{Hilb}^n(\mathcal{A}_r)$, and the relative Gromov-Witten theory of $\mathcal{A}_r \times \mathbb{P}^1$?
• RQ4 Can 2-point extended invariants of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ be expressed in terms of known combinatorial objects like Hurwitz numbers?
• RQ5 Does the change of variables $q = -e^{iu}$ induce a canonical isomorphism between the quantum cohomologies of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ and $\mathrm{Hilb}^n(\mathcal{A}_r)$ that respects quantum product and Poincaré pairing?

Key findings

• The 2-point extended Gromov-Witten invariants of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ are fully determined by equivariant orbifold Poincaré pairings and one-part double Hurwitz numbers, which admit explicit closed formulas.
• The operators of quantum multiplication by divisor classes on $[\mathrm{Sym}^n(\mathcal{A}_r)]$ are completely computed and shown to match those on $\mathrm{Hilb}^n(\mathcal{A}_r)$ under the change of variables $q = -e^{iu}$.
• A linear isomorphism $L$ exists between the equivariant Chen-Ruan cohomology of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ and the equivariant cohomology of $\mathrm{Hilb}^n(\mathcal{A}_r)$, preserving gradings, Poincaré pairings, and quantum product by divisors.
• The Crepant Resolution Conjecture holds for $[\mathrm{Sym}^n(\mathcal{A}_r)]$ and $\mathrm{Hilb}^n(\mathcal{A}_r)$: all extended multi-point functions are preserved under $L$, with $q = -e^{iu}$ as the key transformation.
• The quantum cohomology of $[\mathrm{Sym}^n(\mathcal{A}_r)]$ is shown to be equivalent to the relative Gromov-Witten theory of $\mathcal{A}_r \times \mathbb{P}^1$, suggesting deeper geometric connections.
• The full tetrahedron of equivalences is completed: $[\mathrm{Sym}^n(\mathcal{A}_r)]$-GW, $\mathrm{Hilb}^n(\mathcal{A}_r)$-GW, and relative invariants of $\mathcal{A}_r \times \mathbb{P}^1$ are all isomorphic under $L$ and the change of variables $q = -e^{iu}$.