Mirror symmetry between orbifold curves and cusp singularities with group action

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autorschaft

  • Wolfgang Ebeling
  • Atsushi Takahashi

Organisationseinheiten

Externe Organisationen

  • Osaka University
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Details

OriginalspracheEnglisch
Seiten (von - bis)2240-2270
Seitenumfang31
FachzeitschriftInternational Mathematics Research Notices
Jahrgang2013
Ausgabenummer10
PublikationsstatusVeröffentlicht - 2013

Abstract

We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-Hübsch transpose $(f^T, G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity.

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Mirror symmetry between orbifold curves and cusp singularities with group action. / Ebeling, Wolfgang; Takahashi, Atsushi.
in: International Mathematics Research Notices, Jahrgang 2013, Nr. 10, 2013, S. 2240-2270.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Ebeling, Wolfgang ; Takahashi, Atsushi. / Mirror symmetry between orbifold curves and cusp singularities with group action. in: International Mathematics Research Notices. 2013 ; Jahrgang 2013, Nr. 10. S. 2240-2270.
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abstract = " We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-H{\"u}bsch transpose $(f^T, G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity. ",
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AU - Ebeling, Wolfgang

AU - Takahashi, Atsushi

N1 - Funding information: This work was supported by the DFG-programme SPP 1388 ”Representation Theory” (Eb 102/6-1). The second author was also supported by Grant-in Aid for Scientific Research grant numbers 20360043 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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N2 - We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-Hübsch transpose $(f^T, G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity.

AB - We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-Hübsch transpose $(f^T, G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity.

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UR - https://arxiv.org/abs/1103.5367

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