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Refined curve counting with descendants and quantum mirrors

Patrick Kennedy-Hunt, Qaasim Shafi, Ajith Urundolil Kumaran

TL;DR

This paper develops a refined, q–deformed bridge between higher genus descendant logarithmic Gromov–Witten invariants of log Calabi–Yau surfaces and counts in quantum scattering diagrams, thereby identifying structure constants of the quantum mirror with descendant GW data. The authors formulate and prove a generating-function identity, after the change of variables $q=e^{iu}$, that expresses the descendant invariants $\sum_{g\ge0} N_{g,\vartheta}^{\beta_p}u^{2g}$ in terms of counts of quantum broken lines in the consistent quantum scattering diagram $S(\hat{\mathfrak D}_m)$, with a precise combinatorial factor. The method intertwines the Gross–Siebert program’s degeneration techniques with tropical geometry and a refined tropical correspondence theorem, using diagram factoring and vertex decomposition to relate quantum data to tropical counts. The results generalize the weak Frobenius structure conjecture to the $q$-refined setting for surfaces and illuminate the GW/DT correspondence in a logarithmic, tropical, and quantum framework, offering computational leverage for descendant invariants and deeper insights into quantum mirrors.

Abstract

Given a log Calabi--Yau surface $(Y,D)$, Bousseau has constructed a quantization of the mirror algebra of this pair. We give a formula for structure constants of this quantization in terms of higher genus descendant logarithmic Gromov--Witten invariants of $(Y,D)$. Our result generalises the weak Frobenius structure conjecture for surfaces to the $q$-refined setting, and is proved by relating these invariants to counts of quantum broken lines in the associated quantum scattering diagram.

Refined curve counting with descendants and quantum mirrors

TL;DR

This paper develops a refined, q–deformed bridge between higher genus descendant logarithmic Gromov–Witten invariants of log Calabi–Yau surfaces and counts in quantum scattering diagrams, thereby identifying structure constants of the quantum mirror with descendant GW data. The authors formulate and prove a generating-function identity, after the change of variables , that expresses the descendant invariants in terms of counts of quantum broken lines in the consistent quantum scattering diagram , with a precise combinatorial factor. The method intertwines the Gross–Siebert program’s degeneration techniques with tropical geometry and a refined tropical correspondence theorem, using diagram factoring and vertex decomposition to relate quantum data to tropical counts. The results generalize the weak Frobenius structure conjecture to the -refined setting for surfaces and illuminate the GW/DT correspondence in a logarithmic, tropical, and quantum framework, offering computational leverage for descendant invariants and deeper insights into quantum mirrors.

Abstract

Given a log Calabi--Yau surface , Bousseau has constructed a quantization of the mirror algebra of this pair. We give a formula for structure constants of this quantization in terms of higher genus descendant logarithmic Gromov--Witten invariants of . Our result generalises the weak Frobenius structure conjecture for surfaces to the -refined setting, and is proved by relating these invariants to counts of quantum broken lines in the associated quantum scattering diagram.

Paper Structure

This paper contains 29 sections, 18 theorems, 75 equations, 3 figures.

Table of Contents

  1. Logarithmic Gromov--Witten invariants of $(Y_m,\partial Y_m)$
  2. Quantum broken line counts in quantum scattering diagrams
  3. Future directions
  4. Threefolds
  5. K3 surfaces
  6. Related work
  7. Acknowledgements
  8. Logarithmic Gromov--Witten theory and tropical curves
  9. Looijenga pairs
  10. Logarithmic Gromov--Witten theory
  11. Invariants
  12. Tropical curve counting problems
  13. Tropical curves
  14. Multiplicities
  15. Tropical counting problem
  16. ...and 14 more sections

Key Result

Theorem A

Fix $m=(m_1,\dots,m_n)$ and $\vartheta = (r_1,\dots, r_s)$ tuples of vectors in $\mathbb{Z}^2$. The generating function for the higher genus descendant logarithmic Gromov--Witten invariants is determined by a combinatorial invariant involving counts of quantum broken lines. Precisely, there is an eq after the change of variable $q= e^{iu}$. On the left hand side we sum over all $p=(p_1,\dots,p_n)

Figures (3)

  • Figure 1: A pictorial representation of an example of a summand in \ref{['scattering invariant']} for $m=((1,0),(0,1))$. The black rays are the inital rays of $S(\hat{\mathfrak D}_m)$, the red rays are rays added to make the diagram consistent (see bousseau2020quantum). There are three broken lines $\gamma_1,\gamma_2,\gamma_3$, dotted, in purple. Each broken line ends at the balanced point $Q \in \mathbb{R}^2$, has asymptotic start directions $\vartheta = (r_1,r_2,r_3)$ and final directions $(\varv(\gamma_1),\varv(\gamma_2),\varv(\gamma_3))$.
  • Figure 2: A example of the central fibre $\mathcal{Y}_{m,0}$ and the tropicalisation $\mathcal{Y}_{m,0}^{\mathsf{trop}}$
  • Figure 3: Example of $\Gamma_{k,\underline{g}}^s$ for target as in Figure \ref{['Figure : Target example']}.

Theorems & Definitions (41)

  • Theorem A
  • Corollary B
  • Proposition 1.1.1
  • Proposition 1.2.1
  • proof
  • Definition 1.3.1
  • Definition 1.4.1
  • Proposition 1.5.1
  • proof
  • Remark 1.5.2
  • ...and 31 more