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Two-Sphere Partition Functions and Gromov–Witten Invariants

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Abstract

Many \({\mathcal{N}=(2,2)}\) two-dimensional nonlinear sigma models with Calabi–Yau target spaces admit ultraviolet descriptions as \({\mathcal{N}=(2,2)}\) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories—recently computed via localization by Benini et al. and Doroud et al.—yields the exact Kähler potential on the quantum Kähler moduli space for Calabi–Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov–Witten invariants for any such Calabi–Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α′. We compute these quantities for the quintic and for Rødland’s Pfaffian Calabi–Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi–Yau threefold in \({\mathbb{P}^{7}}\) , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi–Yau is currently known. We derive predictions for its Gromov–Witten invariants and verify that our predictions satisfy nontrivial geometric checks.

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Authors and Affiliations

  1. Bethe Center for Theoretical Physics, Physikalisches Institut, Universität Bonn, 53115, Bonn, Germany

    Hans Jockers

  2. KITP, University of California, Santa Barbara, CA, 93106, USA

    Vijay Kumar

  3. Department of Physics, McGill University, Montreal, QC, Canada

    Joshua M. Lapan

  4. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    David R. Morrison

  5. Department of Physics, University of California, Santa Barbara, CA, 93106, USA

    David R. Morrison

  6. Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Mauricio Romo

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  1. Hans Jockers
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  3. Joshua M. Lapan
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  4. David R. Morrison
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Correspondence to Joshua M. Lapan.

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Communicated by N. A. Nekrasov

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Jockers, H., Kumar, V., Lapan, J.M. et al. Two-Sphere Partition Functions and Gromov–Witten Invariants. Commun. Math. Phys. 325, 1139–1170 (2014). https://doi.org/10.1007/s00220-013-1874-z

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