Higher Genus Gromov-Witten Theory of C^n/Z_n II: Crepant Resolution Correspondence

This paper establishes a higher genus crepant resolution correspondence between the Gromov-Witten theories of the canonical bundle $K\mathbb{P}^{n-1}$ and the orbifold $[\mathbb{C}^n/\mathbb{Z}_n]$ for arbitrary $n \geq 3$ by proving the finite generation of their potentials and constructing an isomorphism between their associated polynomial rings.

The Gist

Imagine you are looking at a complex, crumpled piece of paper. In mathematics, this "paper" represents a shape called a singular variety. It has a sharp, messy point where the geometry breaks down and becomes undefined.

Mathematicians love smooth shapes because they are easier to study. So, they have two main ways to "fix" this crumpled paper:

1. The Orbifold Way ([Cn/Zn]): Instead of smoothing the paper out, they treat the messy point as a special kind of "fold" where the rules of geometry are slightly twisted. They keep the sharp point but wrap it in a mathematical blanket that makes it behave nicely.
2. The Resolution Way (KPn−1): They take a pair of scissors and cut out the messy point, then glue in a smooth, curved surface (like blowing up a balloon) to fill the hole. This creates a completely smooth shape.

In the real world, these two shapes look different. One has a twist; the other has a smooth curve. However, a famous mathematical guess called the Crepant Resolution Conjecture says that if you look at these shapes through the lens of Gromov–Witten theory (a way of counting how many ways strings can wrap around these shapes), they should actually tell the exact same story.

The Problem

For a long time, mathematicians could prove this "same story" idea only for simple cases (like when the shape is 3-dimensional). They struggled to prove it for more complex, higher-dimensional shapes (where $n$ is any number greater than or equal to 3). The math gets incredibly messy when you try to count these string-wrapping patterns in higher dimensions, especially when you look at "higher genus" (which is like counting more complex, multi-looped strings rather than simple circles).

The Solution: A Mathematical Translator

In this paper, Deniz Genlik and Hsian-Hua Tseng act as master translators. They successfully prove that for any dimension $n \ge 3$, the "story" told by the twisted orbifold shape is identical to the "story" told by the smooth resolved shape.

Here is how they did it, using simple analogies:

1. Building a Dictionary (The Polynomial Rings)
To compare the two shapes, the authors first built a specific "dictionary" for each.

• For the twisted shape, they created a ring of functions (a set of mathematical building blocks) where all the counting numbers live.
• For the smooth shape, they built a nearly identical dictionary.
• The Breakthrough: They showed that every single number you can calculate for the smooth shape can be translated into a number for the twisted shape, and vice versa. They proved that the "stories" are generated by the exact same set of rules, just written in slightly different languages.

2. The Givental–Teleman Machine
To handle the complexity of higher dimensions, they used a powerful mathematical tool called the Givental–Teleman classification. Think of this as a high-tech machine that takes a complex, messy shape and breaks it down into simple, fundamental parts (like a deconstructed Lego set).

• The machine produces a "R-matrix" for each shape. This matrix is like a secret code that determines how the strings wrap around the shape.
• The authors had to prove that the secret code for the twisted shape and the secret code for the smooth shape are actually the same code, just shifted by a few mathematical constants.

3. The "Oscillatory" Proof
The hardest part was proving that these secret codes matched. To do this, they looked at oscillatory integrals.

• Imagine a drum skin vibrating. The pattern of the vibration depends on the shape of the drum.
• The authors analyzed the "vibrations" (mathematical integrals) of the smooth shape's mirror image (a concept from mirror symmetry).
• By studying how these vibrations behave at the very edge of infinity (asymptotics), they were able to show that the mathematical "fingerprint" of the smooth shape perfectly matched the fingerprint of the twisted shape.

The Main Result

The paper concludes with a Crepant Resolution Correspondence. This is a precise formula that acts as a translator. If you know the answer for the smooth shape, you can instantly calculate the answer for the twisted shape using this formula, and it will be correct for any dimension $n \ge 3$.

In summary:
The authors took two different ways of fixing a geometric "crumple"—one that keeps the twist and one that smooths it out—and proved that when you count the complex ways strings can wrap around them, the results are mathematically identical. They did this by building a universal dictionary and proving that the secret codes governing both shapes are actually the same, finally solving a puzzle that had only been solved for simple cases before.

Technical Summary

Technical Summary: Higher Genus Gromov–Witten Theory of $[\mathbb{C}^n/\mathbb{Z}_n]$ II: Crepant Resolution Correspondence

Problem Statement
This paper investigates the higher genus Gromov–Witten (GW) theory of two specific targets: the total space $K\mathbb{P}^{n-1}$ of the canonical bundle over the projective space $\mathbb{P}^{n-1}$, and the orbifold $[\mathbb{C}^n/\mathbb{Z}_n]$. The scheme-theoretic quotient $\mathbb{C}^n/\mathbb{Z}_n$ is a singular variety with a unique singular point, while the stack quotient $[\mathbb{C}^n/\mathbb{Z}_n]$ is a smooth Deligne–Mumford stack. The blow-up of the singular point yields $K\mathbb{P}^{n-1}$. Both maps from the stack and the blow-up to the singular variety are birational and crepant.

The central problem addressed is the Crepant Resolution Conjecture, which predicts that the GW theories of $[\mathbb{C}^n/\mathbb{Z}_n]$ and $K\mathbb{P}^{n-1}$ are equivalent. While this equivalence was established for genus 0 (as a special case of results for toric orbifolds) and for specific higher genus cases (notably $n=3$ in [6, 18] and $n=5$ in [17]), this paper aims to prove the correspondence for arbitrary $n \ge 3$. A significant hurdle in extending these results to higher genus is establishing the analytic properties of the GW potentials and demonstrating that they satisfy the necessary finite generation properties within explicit polynomial rings.

Methodology
The authors employ the Givental–Teleman classification of semisimple Cohomological Field Theories (CohFTs). This framework allows the reconstruction of higher genus GW invariants from the genus 0 data (Frobenius manifold structure) via the action of an $R$-matrix and a $T$-vector.

The methodology proceeds through the following steps:

1. Finite Generation: The authors first establish that the GW potentials for both $K\mathbb{P}^{n-1}$ and $[\mathbb{C}^n/\mathbb{Z}_n]$ lie in explicitly constructed polynomial rings. For $K\mathbb{P}^{n-1}$, they construct a ring $F_{K\mathbb{P}^{n-1}}$ generated by specific functions derived from the $I$-function and its derivatives. This parallels their previous work on $[\mathbb{C}^n/\mathbb{Z}_n]$ (Paper [11]).
2. Semisimple Structure: They analyze the genus 0 GW theory to identify the semisimple point and construct the associated Frobenius manifold. This involves calculating the quantum product, the metric in the idempotent basis, and the transition matrix $\Psi$.
3. R-Matrix Identification: Using the Givental–Teleman formalism, the higher genus correspondence is reduced to identifying the $R$-matrices of the two theories. The authors derive "flatness equations" (differential equations) that determine the $R$-matrices.
4. Analytic Comparison: To prove the $R$-matrices are isomorphic under a specific change of variables, the authors compare the solutions to the flatness equations. This requires analyzing the asymptotic expansions of oscillatory integrals associated with the Landau–Ginzburg mirror of $K\mathbb{P}^{n-1}$.
5. Ring Isomorphism: They construct a ring isomorphism $\Upsilon$ between the polynomial rings associated with $K\mathbb{P}^{n-1}$ and $[\mathbb{C}^n/\mathbb{Z}_n]$. This map depends on a choice of an $n$-th root of $-1$, denoted $\rho$.

Key Contributions and Results

• Finite Generation for $K\mathbb{P}^{n-1}$: The paper proves that the GW potential $F_{K\mathbb{P}^{n-1}}^{g,m}$ lies in the ring $F_{K\mathbb{P}^{n-1}} = \mathbb{C}[(L_{K\mathbb{P}^{n-1}})^{\pm 1}][S_{K\mathbb{P}^{n-1}}^n][C_{K\mathbb{P}^{n-1}}^n]$, where the generators are explicit functions of the mirror map variable $q$.
• Crepant Resolution Correspondence (Main Theorem): The authors prove that for $2g-2+m > 0$, the GW generating functions of the two spaces are related by the ring isomorphism $\Upsilon$:
  $$F_{[\mathbb{C}^n/\mathbb{Z}_n]}^{g,m}(\phi_{c_1}, \dots, \phi_{c_m}) = (-1)^{1-g}\rho^{3g-3+m} \Upsilon(F_{K\mathbb{P}^{n-1}}^{g,m}(H_{c_1}, \dots, H_{c_m}))$$
  Here, $\rho$ is an $n$-th root of $-1$ determined by the analytic continuation of the mirror map.
• Generalization of Previous Work: This result generalizes the specific cases of $n=3$ (Lho–Pandharipande [19]) and $n=5$ (Lho [17]) to arbitrary $n \ge 3$.
• Analytic Identity: A crucial technical component is the proof of an identity involving the asymptotic expansions of oscillatory integrals (Lemma 5.8), which equates the constant terms of the $R$-matrix solutions for the two theories. This identity generalizes results found in [19] and [17].

Significance
The paper claims to provide a complete proof of the higher genus Crepant Resolution Correspondence for the family of targets $[\mathbb{C}^n/\mathbb{Z}_n]$ and $K\mathbb{P}^{n-1}$. By utilizing the Givental–Teleman classification, the authors reduce the complex problem of comparing higher genus invariants to the identification of $R$-matrices, which is achieved through a rigorous analysis of flatness equations and asymptotic expansions.

The work demonstrates that the GW theories of these non-compact targets are not only equivalent in genus 0 but are structurally identical in all genera, provided one accounts for the specific analytic continuation and the ring isomorphism $\Upsilon$. This confirms the robustness of the Crepant Resolution Conjecture beyond the compact toric orbifold cases previously established in [5]. The paper does not propose new experimental applications or future directions beyond the scope of this mathematical correspondence, focusing strictly on the structural proof of the equivalence.