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

Part of: Symplectic geometry, contact geometry Projective and enumerative geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  07 January 2020

Pierrick Bousseau
Pierrick Bousseau*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
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Abstract

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.

Keywords

mirror symmetrydeformation quantizationGromov–Witten invariants

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14J33: Mirror symmetry 14J81: Relationships with physics 53D55: Deformation quantization, star products
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 360 - 411
Copyright
© The Author 2020

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Footnotes

1

Current address: Institute for Theoretical Studies, ETH Zurich, 8092 Zurich, Switzerland email pboussea@ethz.ch

This work is supported by EPSRC award 1513338, ‘Counting curves in algebraic geometry’, Imperial College London, and has benefited from the EPRSC [EP/L015234/1], EPSRC Centre for Doctoral Training in Geometry and Number Theory (London School of Geometry and Number Theory), University College London.

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