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. 

Refined count of real oriented rational curves

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

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

TLDR
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

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

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

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

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

  • Travis Mandel
  • Mathematics
    Journal 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

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

References

SHOWING 1-10 OF 141 REFERENCES

Curves in Calabi-Yau threefolds and topological quantum field theory

We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau threefolds. We define relative invariants for the local theory which give rise to a 1+1-dimensional TQFT taking

Gromov-Witten theory of Deligne-Mumford stacks

Given a smooth complex Deligne-Mumford stack ${\cal X}$ with a projective coarse moduli space, we introduce Gromov-Witten invariants of ${\cal X}$ and prove some of their basic properties, including

Mirror symmetry for log Calabi-Yau surfaces I

We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is

Descendant log Gromov-Witten invariants for toric varieties and tropical curves

Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero

Integrality of Gopakumar-Vafa Invariants of Toric Calabi-Yau Threefolds

The Gopakumar–Vafa invariants are numbers defined as certain linear combinations of the Gromov–Witten invariants. We prove that the GV invariants of a toric Calabi–Yau threefold are integers and that

Mirror Symmetry for log Calabi-Yau Surfaces II

We give a canonical compactification of the mirror family to a pair pY,Dq where Y is a rational surface, D is an anti-canonical cycle of rational curves and Y zD is the minimal resolution of an

Refined tropical curve counts and canonical bases for quantum cluster algebras

We express the quantum and classical versions of the Gross-Hacking-Keel-Kontsevich theta bases for cluster algebras in terms of certain descendant tropical Gromov-Witten invariants (with

Refined GW/Kronecker correspondence

Gromov–Witten invariants of weighted projective planes and Euler characteristics of moduli spaces of representations of bipartite quivers are related via the tropical vertex, a group of formal

Tropical refined curve counting via motivic integration

We propose a geometric interpretation of Block and G\"ottsche's refined tropical curve counting invariants in terms of virtual $\chi_{-y}$-specializations of motivic measures of semialgebraic sets in

Localization of virtual classes

We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of
...