Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting
File(s)
Author(s)
Bousseau, Pierrick
Type
Thesis
Abstract
We present three results, at the intersection of tropical geometry, enumerative geometry, mirror symmetry and non-commutative algebra.
1. A correspondence between Block-Göttsche q-refined tropical curve counting and higher genus log Gromov-Witten theory of toric surfaces.
2. A correspondence between q-refined two-dimensional Kontsevich-Soibelman scattering diagrams and higher genus log Gromov-Witten theory of log Calabi-Yau surfaces.
3. A q-deformation of the Gross-Hacking-Keel mirror construction, producing a deformation quantization with canonical basis for the Gross-Hacking-Keel families of log
Calabi-Yau surfaces.
These results are logically dependent: the proof of the third result relies on the second, whose proof itself relies on the first. Nevertheless, each of them is of independent interest.
1. A correspondence between Block-Göttsche q-refined tropical curve counting and higher genus log Gromov-Witten theory of toric surfaces.
2. A correspondence between q-refined two-dimensional Kontsevich-Soibelman scattering diagrams and higher genus log Gromov-Witten theory of log Calabi-Yau surfaces.
3. A q-deformation of the Gross-Hacking-Keel mirror construction, producing a deformation quantization with canonical basis for the Gross-Hacking-Keel families of log
Calabi-Yau surfaces.
These results are logically dependent: the proof of the third result relies on the second, whose proof itself relies on the first. Nevertheless, each of them is of independent interest.
Version
Open Access
Date Issued
2018-06
Date Awarded
2018-09
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Thomas, Richard
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
1513338
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)