# Quantum mirrors of log Calabi–Yau surfaces and higher-genus curve counting

@article{Bousseau2020QuantumMO, title={Quantum mirrors of log Calabi–Yau surfaces and higher-genus curve counting}, author={Pierrick Bousseau}, journal={Compositio Mathematica}, year={2020}, volume={156}, pages={360 - 411} }

Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher-genus log Gromov–Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding non-commutative algebras of functions.

## 17 Citations

### Refined count of real oriented rational curves

- Mathematics
- 2021

We introduce a quantum index for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen 2-form. We then…

### Log Calabi-Yau surfaces and Jeffrey-Kirwan residues

- Mathematics
- 2021

We use the mirror construction of Gross, Hacking and Keel in order to prove a version, for a class of log Calabi-Yau surfaces, of the general expectation appearing in physics, in the context of…

### Tropical refined curve counting from higher genera and lambda classes

- MathematicsInventiones mathematicae
- 2019

It is shown that the result is a generating series of higher genus log Gromov–Witten invariants with insertion of a lambda class, which gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.

### On quasi-tame Looijenga pairs

- Mathematics
- 2022

We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact log Gromov–Witten invariants of Looijenga…

### The quantum tropical vertex

- Mathematics
- 2018

Gross-Pandharipande-Siebert have shown that the 2-dimensional Kontsevich-Soibelman scattering diagrams compute certain genus zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that…

### On an example of quiver DT/relative GW correspondence

- Mathematics
- 2018

We explain and generalize a recent result of Reineke-Weist by showing how to reduce it to the Gromov-Witten/Kronecker correspondence by a degeneration and blow-up. We also refine the result by…

### The Quantum Mirror to the Quartic del Pezzo Surface

- Mathematics
- 2021

A log Calabi–Yau surface (X,D) is given by a smooth projective surface X , together with an anti-canonical cycle of rational curves D ⊂ X . The homogeneous coordinate ring of the mirror to such a…

### Scattering diagrams, theta functions, and refined tropical curve counts

- MathematicsJournal of the London Mathematical Society
- 2021

In the Gross–Siebert mirror symmetry program, certain enumerations of tropical disks are encoded in combinatorial objects called scattering diagrams and broken lines. These, in turn, are used to…

### On an Example of Quiver Donaldson–Thomas/Relative Gromov–Witten Correspondence

- Mathematics
- 2020

We explain and generalize a recent example of quiver Donaldson–Thomas/relative Gromov–Witten correspondence due to Reineke–Weist by showing how to reduce it to the Gromov–Witten/Kronecker…

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