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On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities AbstractIn earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov–Witten invariants of X, coincides with the classical period of its mirror partner f.In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with13$\begin{array}{}\frac{1}{3}\end{array} $(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

Oneto, Alessandro; Petracci, Andrea
Advances in Geometry , Volume 18 (3): 34 – Jul 26, 2018
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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1615-7168
eISSN
1615-7168
DOI
10.1515/advgeom-2017-0048
Publisher site
See Article on Publisher Site

Abstract

AbstractIn earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov–Witten invariants of X, coincides with the classical period of its mirror partner f.In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with13$\begin{array}{}\frac{1}{3}\end{array} $(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.

Journal

Advances in Geometry – de Gruyter

Published: Jul 26, 2018

References

  • Del Pezzo surfaces with 1 (1, 1) points