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Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

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Abstract

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold \({\mathcal{X}}\) and that of its toric crepant resolution Y coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Ruan’s original CRC (Gromov-Witten theory of spin curves and orbifolds, contemp math, Amer. Math. Soc., Providence, RI, pp 117–126, 2006). We prove the open CRC for the weighted projective spaces \({\mathcal{X} = \mathbb{P}(1,\ldots,1, n)}\) using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Kwokwai Chan

  2. Department of Mathematical Sciences, Research institute of Mathematics, Seoul National University, San 56-1, Shinrimdong, Gwanakgu, Seoul, 47907, Korea

    Cheol-Hyun Cho

  3. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA, 02138, USA

    Siu-Cheong Lau

  4. Department of Mathematics, Ohio State University, 100 Math Tower, 231 West 18th Ave., Columbus, OH, 43210, USA

    Hsian-Hua Tseng

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  1. Kwokwai Chan
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  2. Cheol-Hyun Cho
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  3. Siu-Cheong Lau
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  4. Hsian-Hua Tseng
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Correspondence to Cheol-Hyun Cho.

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Communicated by N. A. Nekrasov

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Chan, K., Cho, CH., Lau, SC. et al. Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds. Commun. Math. Phys. 328, 83–130 (2014). https://doi.org/10.1007/s00220-014-1948-6

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