Remodeling Conjecture with Descendants

TL;DR

The paper proves the Remodeling Conjecture with descendants for semi-projective toric Calabi-Yau 3-orbifolds by establishing a canonical link between equivariant K-theory with support, integral structures, and the equivariant mirror curve. It develops genus-zero and all-genus correspondences: (i) a T'-equivariant Gamma conjecture linking central charges to oscillatory integrals on the mirror, (ii) Hosono's conjecture in the non-equivariant limit via a K^c–H_3(Ĉ_q) isomorphism, and (iii) a full all-genus descendant mirror symmetry via topological recursion on the mirror curve, identifying A-model descendant Gromov-Witten invariants with B-model ω_{g,n} integrals along mirror cycles through a canonical isomorphism mir^+_{T'}. The work introduces and analyzes K-theory with bounded below/above support to provide robust integral structures, constructs mirror cycles for generators of K^+_{T'}(𝒳) and shows their central charges reproduce GW data under the mirror map, and extends BKMP remodeling to the descendant setting for toric CY3 orbifolds. Collectively, these results illuminate the deep interplay between A- and B-models in non-compact, equivariant Calabi–Yau geometries and yield tools for future exploration of crepant transformations and higher-genus dualities.

Abstract

We formulate and prove the Remodeling Conjecture with descendants, which is a version of all-genus equivariant descendant mirror symmetry for semi-projective toric Calabi-Yau 3-orbifolds with integral structures. We construct an isomorphism between the $K$-group of equivariant coherent sheaves on the toric Calabi-Yau 3-orbifold with support bounded in a direction and a certain integral relative first homology group of the equivariant mirror curve. Under this isomorphism, we prove the equivariant mirror symmetric Gamma conjecture which equates quantum cohomology central charges of coherent sheaves and oscillatory integrals along corresponding relative 1-cycles. As a consequence in the non-equivariant setting, we prove a conjecture of Hosono which equates central charges of compactly supported coherent sheaves and period integrals of integral 3-cycles on the Hori-Vafa mirror 3-fold. Furthermore, we establish a correspondence between all-genus equivariant descendant Gromov-Witten invariants with $K$-theoretic framings and oscillatory integrals (Laplace transforms) of the Chekhov-Eynard-Orantin topological recursion invariants along relative 1-cycles on the equivariant mirror curve.

Figures (4)

• Figure 1: Mirror curve $C_ {\sigma}$ and cycles $\gamma_\chi$, $\gamma'_\chi$. The punctures are illustrated by circles, and the curve may have non-zero genus (not illustrated).
• Figure 2: Patching cycles for the mirror of a toric divisor. The paths $\gamma^\circ_{\mathbf{f}_i}$ and $\gamma^\circ_{\mathbf{f}_{i+1}}$ connect to the same component of $\widetilde{C}^\circ_{\tau_{i+1}}$. This component is homeomorphic to $\mathbb{R} \times [0,1]$. Passing down to the component of $C^\circ_{\tau_{i+1}}$ (homeomorphic to $S^1\times [0,1]$) this may have a non-trivial winding (dotted magenta part).
• Figure 3: Patching cycles for the mirror of a compact toric curve. The paths $\gamma'^\circ_{ {\sigma} _1}$ and $\gamma'^\circ_{ {\sigma} _{2}}$ connect to the same component of $\widetilde{C}^\circ_{\tau}$. This component is homeomorphic to $\mathbb{R} \times [0,1]$. Passing down to the component of $C^\circ_{\tau}$ (homeomorphic to $S^1\times [0,1]$) this may have a non-trivial winding (dotted cyan part).
• Figure 4: Wall-and-chamber structure on $N'_{\mathbb{R}}$ for $\mathcal{X} = K_{\mathbb{P}^2}$. The walls are dual to the images of the three non-compact toric curves under the moment map $\mu_{\mathbb{T}'_{\mathbb{R}}}: \mathcal{X} \to M'_{\mathbb{R}}$. The blue parts on the left illustrate the images of $\mathcal{X}^+$ under $\mu_{\mathbb{T}'_{\mathbb{R}}}$ for the different chambers. The orange parts on the right highlight the punctures (on $C_q$) contained in $\{\mathrm{Re} (\hat{x}) \gg 0\}$.

Theorems & Definitions (85)

• Theorem 1.1: = Theorem \ref{['thm:EquivMir']}) (Genus-zero descendant mirror theorem
• Theorem 1.2: = Theorem \ref{['thm:Kc3Mir']}) (Hosono's conjecture
• Remark 1.3
• Theorem 1.4: = Theorem \ref{['thm:All-genus-mirror']}) (All-genus descendant mirror theorem
• Theorem 3.2
• Theorem 3.3: Zong Zong15
• Theorem 3.5
• Definition 3.6
• Lemma 3.7
• Definition 3.8