[Paper Review] Higher genus quasimap wall-crossing for semi-positive targets

Ionuţ Ciocan-Fontanine, Bumsig Kim|arXiv (Cornell University)|Aug 29, 2013

Algebraic Geometry and Number Theory14 references20 citations

TL;DR

This paper establishes higher genus wall-crossing formulas for quasimap invariants of semi-positive GIT quotients, extending genus-zero results to higher genera. It proves the conjectured relation between Gromov-Witten and quasimap potentials via the $J$-function and its $1/z$-expansion, with a complete proof for semi-positive toric varieties and local Calabi-Yau targets using localization and $T$-equivariant techniques.

ABSTRACT

In previous work (arXiv:1304.7056) we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies to local Calabi-Yau's associated to some non-abelian quotients.

Motivation & Objective

• To extend genus-zero quasimap wall-crossing formulas to higher genus in the context of semi-positive GIT quotients.
• To prove the conjectured wall-crossing formula relating Gromov-Witten and quasimap potentials for higher genus invariants.
• To establish the formula for semi-positive toric varieties and local Calabi-Yau targets via localization and $T$-equivariant methods.
• To verify that the wall-crossing transformation is governed by the $1/z$-expansion of the small $J$-function.

Proposed method

• Uses the $J$-function $J^\varepsilon(q,t,z)$ and its small version $J^\varepsilon_{sm}(q,z)$ to relate quasimap and Gromov-Witten invariants.
• Employs the $1/z$-expansion of the small $J$-function: $J^\varepsilon_{sm}(q,z) = J^\varepsilon_0(q)\mathbbm{1} + J^\varepsilon_1(q)/z + O(1/z^2)$.
• Applies localization techniques on the moduli space of $\varepsilon$-stable quasimaps to compute vertex contributions and match $J$-functions.
• Uses the small $I$-function $I_{sm}(q,z)$ as the $\varepsilon \to 0^+$ limit of $J^\varepsilon_{sm}$, with $I_0 = 1$ under semi-positivity.
• Relies on the $T$-equivariant structure of the target to ensure isolated fixed points and 1-dimensional orbits, enabling precise vertex computations.
• Verifies the wall-crossing formula by showing that the transformed potential $F^\varepsilon_g$ matches the Gromov-Witten potential $F^\infty_g$ via the $J$-function coefficients.

Experimental results

Research questions

• RQ1 How do quasimap invariants at higher genus relate to Gromov-Witten invariants under wall-crossing in semi-positive targets?
• RQ2 Can the genus-zero wall-crossing formula, previously proven via localization, be extended to higher genus?
• RQ3 What is the precise transformation law between quasimap and Gromov-Witten potentials in higher genus for semi-positive GIT quotients?
• RQ4 Does the $1/z$-expansion of the $J$-function govern the wall-crossing transformation in higher genus?
• RQ5 Under what conditions does the wall-crossing formula hold uniformly across all stability parameters $\varepsilon$?

Key findings

• The wall-crossing formula $ (J^\varepsilon_0)^{2g-2} F^\varepsilon_g(\mathbf{t}) = F^\infty_g\left( \frac{\mathbf{t} + J^\varepsilon_1}{J^\varepsilon_0} \right) $ holds for all $\varepsilon \geq 0^+$ in the semi-positive case.
• The formula is proven for semi-positive toric varieties and local Calabi-Yau targets via localization and $T$-equivariant techniques.
• For $g=1$, a correction term $\frac{1}{24}\chi_{\text{top}}(X)\log J^\varepsilon_0$ is required due to the failure of the dilaton equation on $\overline{M}_{1,1}$.
• The small $I$-function satisfies $I_0 = 1$ for semi-positive triples, which simplifies the wall-crossing transformation.
• The wall-crossing formula is invariant under the transformation $\mathbf{t}(\psi) \mapsto J^\varepsilon_0 \mathbf{t}(\psi) - J^\varepsilon_1$, ensuring consistency across different $\varepsilon$.
• The result extends to local Grassmannians and flag varieties, where the $I$-function is explicitly computed and the wall-crossing holds due to uniform vertex contributions.