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Mirror theorems for root stacks and relative pairs

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Abstract

Given a smooth projective variety X with a smooth nef divisor D and a positive integer r, we construct an I-function, an explicit slice of Givental’s Lagrangian cone, for Gromov–Witten theory of the root stack \(X_{D,r}\). As an application, we also obtain an I-function for relative Gromov–Witten theory following the relation between relative and orbifold Gromov–Witten invariants.

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Notes

  1. Following the custom in mirror theorems, the term “I-function” refers to an explicitly constructed slice of Givental’s Lagrangian cone.

  2. We need to require the cohomology classes at orbifold marking to be in \(i^*H^*(X)\) instead of \(H^*(D)\), because we apply quantum Lefschetz.

  3. The basis \(\{\phi _\alpha \}\) is a basis of the cohomology ring pullback from the cohomological ring of \(X_{D,r}\) to \(D_r\).

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Acknowledgements

F.Y. would like to thank Qile Chen, Charles Doran and Melissa Liu for helpful discussions. H.F. is supported by Grant ERC-2012-AdG-320368-MCSK and SwissMAP. H.-H. T. is supported in part by NSF Grant DMS-1506551. F.Y. is supported by a postdoctoral fellowship funded by NSERC and Department of Mathematical Sciences at the University of Alberta.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Honglu Fan

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

    Hsian-Hua Tseng

  3. Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Fenglong You

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  1. Honglu Fan
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  2. Hsian-Hua Tseng
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Correspondence to Fenglong You.

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Fan, H., Tseng, HH. & You, F. Mirror theorems for root stacks and relative pairs. Sel. Math. New Ser. 25, 54 (2019). https://doi.org/10.1007/s00029-019-0501-z

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