Communications in Number Theory and Physics

Volume 7 (2013)

Number 4

Integrality of relative BPS state counts of toric del Pezzo surfaces

Pages: 671 – 687



Michel van Garrel (KIAS, Seoul, Korea)

Tony W. H. Wong (Department of Mathematics, Kutztown University of Pennsylvania, Kutztown, Penn., U.S.A.)

Gjergji Zaimi (Department of Mathematics, California Institute of Technology, Pasadena, Calif., U.S.A.)


Relative Bogomolny-Prasad-Sommerfield (BPS) state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert in [4] and conjectured by the authors to be integers. For toric del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng in [15]; and more generally for toric Calabi-Yau three-folds by Konishi in [8].

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