[Paper Review] Root Systems and the Quantum Cohomology of ADE resolutions

Jim Bryan, Amin Gholampour|ArXiv.org|Jul 9, 2007

Algebraic Geometry and Number Theory6 references13 citations

TL;DR

This paper computes the ${\mathbb{C}}^{*}$-equivariant quantum cohomology ring of the minimal resolution $Y$ of the DuVal singularity ${\mathbb{C}}^2/G$ for finite $G \subset SU(2)$, expressing the quantum product in terms of the ADE root system associated to $G$. The key result is a closed formula for the quantum product using the root system's structure, generalizing to non-simply laced root systems and predicting the orbifold Gromov-Witten potential via the Crepant Resolution Conjecture.

ABSTRACT

We compute the C*-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C^2/G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov-Witten potential of [C^2/G].

Motivation & Objective

• To establish a quantum extension of the classical McKay correspondence by describing the quantum cohomology of $Y$, the minimal resolution of ${\mathbb{C}}^2/G$, in terms of the ADE root system associated to $G$.
• To compute the ${\mathbb{C}}^{*}$-equivariant quantum cohomology ring of $Y$ using genus 0 Gromov-Witten invariants and root system data.
• To generalize the resulting Frobenius algebra structure to non-simply laced root systems, providing an $n$-parameter family of algebra structures on the affine root lattice.
• To use the Crepant Resolution Conjecture to predict the orbifold Gromov-Witten potential of the stack $[{\/mathbb{C}}^2/G]$.

Proposed method

• The quantum product is expressed via a formula involving the Cartan matrix and quantum parameters $q_i$, with the structure constants given by $\left\langle v,\beta\right\rangle\left\langle w,\beta\right\rangle t\frac{1+q^{\beta}}{1-q^{\beta}}\beta$ over positive roots $\beta$.
• The pairing on the quantum cohomology is extended to a ${\mathbb{Q}}[t,t^{-1}][[q_1,\ldots,q_n]]$-valued pairing, with $\left\langle 1,1\right\rangle = -1/(t^2|G|)$.
• The computation of genus 0 equivariant Gromov-Witten invariants is achieved by relating $Y$ to a threefold $W$ via deformation families and applying the method of Bryan, Katz, and Leung.
• Degree-zero invariants are computed using localization techniques, and the potential function is derived via triple derivatives of the generating function.
• The Weyl group action is shown to preserve the Frobenius algebra structure, and associativity is proven in the non-simply laced case by reduction to the simply laced case.
• A conjectural formula for the orbifold Gromov-Witten potential of $[{\/mathbb{C}}^2/G]$ is derived by specializing quantum parameters to roots of unity and relating to trigonometric functions via $H(u) = \frac{1}{2}\tan(-u/2)$.

Experimental results

Research questions

• RQ1 How can the ${\mathbb{C}}^{*}$-equivariant quantum cohomology of the minimal resolution $Y$ of ${\mathbb{C}}^2/G$ be expressed in terms of the ADE root system associated to $G$?
• RQ2 What is the algebraic structure of the quantum cohomology ring, and how does it generalize to non-simply laced root systems?
• RQ3 How does the Weyl group act on the quantum cohomology, and does it preserve the Frobenius algebra structure?
• RQ4 Can the Crepant Resolution Conjecture be used to predict the orbifold Gromov-Witten potential of $[{\/mathbb{C}}^2/G]$ based on the resolution's invariants?

Key findings

• The quantum product of two classes $v,w \in H^2(Y,\mathbb{Z})$ is given by $v \star w = -t^2|G|\left\langle v,w\right\rangle + \sum_{\beta \in R^+} \left\langle v,\beta\right\rangle\left\langle w,\beta\right\rangle t\frac{1+q^{\beta}}{1-q^{\beta}}\beta$, with $q^\beta = \prod q_i^{b_i}$ for $\beta = \sum b_i \alpha_i$.
• The quantum cohomology ring is a Frobenius algebra over ${\mathbb{Q}}[t,t^{-1}][[q_1,\ldots,q_n]]$, satisfying $\left\langle v\star w,u\right\rangle = \left\langle v,w\star u\right\rangle$.
• The algebra structure is generalized to non-simply laced root systems via reduction to the simply laced case, and associativity is proven in this setting.
• The Weyl group acts as an automorphism on the Frobenius algebra, preserving the quantum product structure.
• A conjectural formula for the orbifold Gromov-Witten potential of $[{\/mathbb{C}}^2/G]$ is derived as $F_{\mathcal{X}} = 2t \sum_{\beta \in R^+} h(Q_\beta)$, where $h'''(u) = \frac{1}{2}\tan(-u/2)$ and $Q_\beta = \pi + \sum_k \frac{b_k}{|G|}(2\pi n_k + \sum_g \sqrt{2 - \chi_V(g)} \bar{\chi}_k(g) x_{[g]})$.
• The conjecture is verified for $G = \mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_4$, and recently for all $\mathbb{Z}_n$, and is consistent with monodromy constraints and subgroup reduction relations.