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Two-Sphere Partition Functions and Gromov-Witten Invariants

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

TL;DR

The authors propose that the exact two-sphere partition function Z_{S^2} of UV N=(2,2) GLSMs flows to Calabi–Yau threefolds computes the exact Kähler potential on the quantum Kähler moduli space, enabling direct extraction of genus-zero Gromov–Witten invariants without mirror symmetry. They develop a concrete procedure to obtain GW data from Z_{S^2}, and validate it on the quintic and Rdland’s Pfaffian Calabi–Yau, finding perfect agreement with known results. Extending to a determinantal Gulliksen–Negård Calabi–Yau in P^7, they produce new predictions for GW invariants and verify nontrivial geometric checks, including line counts and a consistent extremal transition to a related manifold. The work suggests a robust, mirror-free route to quantum-corrected moduli space data and highlights a potential tt^*-fusion interpretation of the partition function.

Abstract

Many N=(2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as 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 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.

Two-Sphere Partition Functions and Gromov-Witten Invariants

TL;DR

The authors propose that the exact two-sphere partition function Z_{S^2} of UV N=(2,2) GLSMs flows to Calabi–Yau threefolds computes the exact Kähler potential on the quantum Kähler moduli space, enabling direct extraction of genus-zero Gromov–Witten invariants without mirror symmetry. They develop a concrete procedure to obtain GW data from Z_{S^2}, and validate it on the quintic and Rdland’s Pfaffian Calabi–Yau, finding perfect agreement with known results. Extending to a determinantal Gulliksen–Negård Calabi–Yau in P^7, they produce new predictions for GW invariants and verify nontrivial geometric checks, including line counts and a consistent extremal transition to a related manifold. The work suggests a robust, mirror-free route to quantum-corrected moduli space data and highlights a potential tt^*-fusion interpretation of the partition function.

Abstract

Many N=(2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as 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 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.

Paper Structure

This paper contains 14 sections, 62 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Review
  3. Local special Kähler manifolds
  4. The complex structure moduli space $\mathcal{M}_{CS}$
  5. The quantum Kähler moduli space $\mathcal{M}_{\text{K\"ahler}}$
  6. The partition function $Z_{S^2}$
  7. The conjecture $Z_{S^2} = e^{-K}$
  8. Interpretation via $tt^*$ equations
  9. Extracting Gromov--Witten invariants from the partition function
  10. Examples
  11. The quintic threefold
  12. Rødland's Pfaffian Calabi--Yau threefold
  13. The determinantal Gulliksen-Negård Calabi--Yau threefold
  14. Partition function and instanton numbers

Figures (1)

  • Figure 1: The classical GLSM moduli space, as a function of the FI parameters $(r_0,r_1)$, of the GLSM that describes the GN Calabi--Yau threefold. The grey shading represents regions of the FI parameter space where we have a classical geometric description. The solid black lines indicate boundaries of the respective geometric phases, along which classical Coulomb branches emerge. The I-II phase boundary has slope $-\frac{1}{2}$.