[Paper Review] The Crepant Resolution Conjecture for 3-dimensional flags modulo an involution

W. D. Gillam|ArXiv.org|Aug 6, 2007

Algebraic Geometry and Number Theory9 references4 citations

TL;DR

This paper verifies the Crepant Resolution Conjecture (CRC) for the orbifold $[F/\mathbb{Z}_2]$, where $F$ is the flag manifold of $\mathbb{C}^3$, by showing that the genus-zero Gromov-Witten potential of the orbifold matches that of its crepant resolution $Y$ after a linear change of variables, setting quantum parameters to $-1$, and analytically continuing coefficients. The key result is an explicit agreement of potential functions under these transformations, confirmed via computation of 3-point invariants and use of WDVV and the Divisor Axiom.

ABSTRACT

After fixing a non-degenerate bilinear form on a vector space V we define an involution of the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov-Witten potential function of [F / Z_2] agrees (up to unstable terms) with the genus zero Gromov-Witten potential function of a crepant resolution Y of the quotient scheme F / Z_2, after setting a quantum parameter to -1, making a linear change of variables, and analytically continuing coefficients. The crepant resolution Y (a hypersurface in the Hilbert scheme Hilb^2 P^2) is the projectivization of a novel rank 2 vector bundle over P^2.

Motivation & Objective

• To verify the Crepant Resolution Conjecture (CRC) for the orbifold $[F/\mathbb{Z}_2]$, where $F$ is the flag manifold of $\mathbb{C}^3$ under a $\mathbb{Z}_2$-action induced by a non-degenerate bilinear form.
• To compute the genus-zero Gromov-Witten potential of the orbifold $[F/\mathbb{Z}_2]$ using Chen-Ruan cohomology and orbifold stable maps.
• To construct and describe the crepant resolution $Y$ of $F/\mathbb{Z}_2$ as a $\mathbb{P}^1$-bundle over $\mathbb{P}^2$ inside the Hilbert scheme $\mathrm{Hilb}^2\mathbb{P}^2$.
• To establish isomorphism between the quantum cohomology rings of $Y$ and $[F/\mathbb{Z}_2]$ via a linear change of variables and parameter specialization.

Proposed method

• Computes the Chen-Ruan orbifold cohomology ring of $[F/\mathbb{Z}_2]$ using the $\mathbb{Z}_2$-invariant cohomology and fixed locus cohomology of the flag manifold.
• Explicitly computes degree 0 and 3-point Gromov-Witten invariants for the orbifold using moduli spaces of stable maps and associativity of the quantum product.
• Uses the WDVV equation and Divisor Axiom to determine higher-genus invariants from lower-degree data.
• Describes the crepant resolution $Y$ as a hypersurface in $\mathrm{Hilb}^2\mathbb{P}^2$, realizing it as a projectivization of a rank 2 vector bundle over $\mathbb{P}^2$.
• Derives a linear change of variables between the quantum parameters of $Y$ and $[F/\mathbb{Z}_2]$ by matching the quantum product structure via the transpose of the quantum product matrix.
• Performs analytic continuation and sets quantum parameters $q_1 = -1$ to match the orbifold potential with the resolution potential.

Experimental results

Research questions

• RQ1 Does the Crepant Resolution Conjecture hold for the orbifold $[F/\mathbb{Z}_2]$, where $F$ is the flag manifold of $\mathbb{C}^3$?
• RQ2 What is the explicit form of the genus-zero Gromov-Witten potential for the orbifold $[F/\mathbb{Z}_2]$ under the $\mathbb{Z}_2$-action induced by a bilinear form?
• RQ3 How does the quantum cohomology ring of the crepant resolution $Y$ of $F/\mathbb{Z}_2$ compare to that of the orbifold $[F/\mathbb{Z}_2]$?
• RQ4 What linear transformation of quantum parameters and variables equates the Gromov-Witten potential of $Y$ with that of $[F/\mathbb{Z}_2]$ after setting $q_1 = -1$?

Key findings

• The genus-zero Gromov-Witten potential of the crepant resolution $Y$ matches that of the orbifold $[F/\mathbb{Z}_2]$ after a linear change of variables and setting $q_1 = -1$, confirming the CRC for this case.
• The potential functions of $Y$ and $[F/\mathbb{Z}_2]$ are equal under the transformation $t_0 \mapsto s_0$, $t_1 \mapsto s_1 + i s_2$, $t_2 \mapsto -i s_2$, $t_3 \mapsto -6s_3 - \frac{3i}{2}s_4$, $t_4 \mapsto 3s_3 + i s_4$, $t_5 \mapsto 3s_5$, with $q_1 = -1$ and $q_2$ absorbed via analytic continuation.
• The 3-point invariants of both $Y$ and $[F/\mathbb{Z}_2]$ are computed explicitly, and higher invariants are determined via the WDVV equation and Divisor Axiom.
• The resolution $Y$ is realized as a $\mathbb{P}^1$-bundle over $\mathbb{P}^2$, specifically a hypersurface in $\mathrm{Hilb}^2\mathbb{P}^2$, and is isomorphic to the projectivization of a novel rank 2 vector bundle on $\mathbb{P}^2$.
• The coefficient of $t_1^n$ in $F^Y$ under $q_2 = 0$ is determined by classical invariants and matches the orbifold potential under the change of variables, confirming agreement in the $s_2$-dependence.
• The third derivatives of the potentials with respect to $s_2$ agree after analytic continuation to $q_1 = -1$, matching the known result for $[\mathbb{C}^2/\mathbb{Z}_2]$, with a factor of 6 and sign adjustment accounted for.