Communications in Mathematical Physics

, Volume 325, Issue 3, pp 1139–1170 | Cite as

Two-Sphere Partition Functions and Gromov–Witten Invariants

  • Hans Jockers
  • Vijay Kumar
  • Joshua M. Lapan
  • David R. Morrison
  • Mauricio Romo


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.


Partition Function Modulus Space Complete Intersection Toric Variety Gauge Linear Sigma Model 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hans Jockers
    • 1
  • Vijay Kumar
    • 2
  • Joshua M. Lapan
    • 3
  • David R. Morrison
    • 4
    • 5
  • Mauricio Romo
    • 6
  1. 1.Bethe Center for Theoretical Physics, Physikalisches InstitutUniversität BonnBonnGermany
  2. 2.KITPUniversity of CaliforniaSanta BarbaraUSA
  3. 3.Department of PhysicsMcGill UniversityMontrealCanada
  4. 4.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  5. 5.Department of PhysicsUniversity of CaliforniaSanta BarbaraUSA
  6. 6.Kavli IPMU (WPI)The University of TokyoKashiwaJapan

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