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Exact Kahler Potential for Calabi-Yau Fourfolds

Yoshinori Honma, Masahide Manabe

TL;DR

This work extends the exact Kähler potential framework to Calabi–Yau fourfolds by leveraging the two-sphere partition function $Z_{ ext{GLSM}}$ of 2D ${\ m N}=(2,2)$ GLSMs, following the approach of Jockers et al. The authors conjecture an explicit formula for the quantum-corrected Kähler potential $e^{-K(z,\bar z)}$ near the large-radius point, expressed through classical quadruple intersections $\kappa_{ijk\ell}$ and quantum corrections encoded in generating functions tied to genus-zero Gromov–Witten invariants; flat coordinates $t^\ell$ are obtained from GLSM data via a mirror-like construction. They validate the conjecture by computing GLSM partition functions for multiple compact CY4s (e.g., sextic, two-moduli quintic fibration, determinantal sextic, Grassmannian complete intersections) and several local toric fourfolds, extracting GW invariants $n_{\beta}$ and showing exact agreement with mirror symmetry predictions. The work also develops a local toric analogue of the GLSM–Kähler potential correspondence, using regulator holomorphic functions to handle non-compact divergences. Overall, the results indicate a powerful, dimension-agnostic framework to access non-perturbative invariants in higher-dimensional Calabi–Yau manifolds with potential applications to non-perturbative aspects of F-theory compactifications.

Abstract

We study quantum Kahler moduli space of Calabi-Yau fourfolds. Our analysis is based on the recent work by Jockers et al. which gives a novel method to compute the Kahler potential on the quantum Kahler moduli space of Calabi-Yau manifold. In contrast to Calabi-Yau threefold, the quantum nature of higher dimensional Calabi-Yau manifold is yet to be fully elucidated. In this paper we focus on the Calabi-Yau fourfold. In particular, we conjecture the explicit form of the quantum-corrected Kahler potential. We also compute the genus zero Gromov-Witten invariants and test our conjecture by comparing the results with predictions from mirror symmetry. Local toric Calabi-Yau varieties are also discussed.

Exact Kahler Potential for Calabi-Yau Fourfolds

TL;DR

This work extends the exact Kähler potential framework to Calabi–Yau fourfolds by leveraging the two-sphere partition function of 2D GLSMs, following the approach of Jockers et al. The authors conjecture an explicit formula for the quantum-corrected Kähler potential near the large-radius point, expressed through classical quadruple intersections and quantum corrections encoded in generating functions tied to genus-zero Gromov–Witten invariants; flat coordinates are obtained from GLSM data via a mirror-like construction. They validate the conjecture by computing GLSM partition functions for multiple compact CY4s (e.g., sextic, two-moduli quintic fibration, determinantal sextic, Grassmannian complete intersections) and several local toric fourfolds, extracting GW invariants and showing exact agreement with mirror symmetry predictions. The work also develops a local toric analogue of the GLSM–Kähler potential correspondence, using regulator holomorphic functions to handle non-compact divergences. Overall, the results indicate a powerful, dimension-agnostic framework to access non-perturbative invariants in higher-dimensional Calabi–Yau manifolds with potential applications to non-perturbative aspects of F-theory compactifications.

Abstract

We study quantum Kahler moduli space of Calabi-Yau fourfolds. Our analysis is based on the recent work by Jockers et al. which gives a novel method to compute the Kahler potential on the quantum Kahler moduli space of Calabi-Yau manifold. In contrast to Calabi-Yau threefold, the quantum nature of higher dimensional Calabi-Yau manifold is yet to be fully elucidated. In this paper we focus on the Calabi-Yau fourfold. In particular, we conjecture the explicit form of the quantum-corrected Kahler potential. We also compute the genus zero Gromov-Witten invariants and test our conjecture by comparing the results with predictions from mirror symmetry. Local toric Calabi-Yau varieties are also discussed.

Paper Structure

This paper contains 23 sections, 115 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Some known results
  3. Moduli spaces and mirror symmetry
  4. Kähler potential and ${\cal N}=(2,2)$ GLSM
  5. Kähler potential for Calabi-Yau fourfolds
  6. Examples
  7. Sextic fourfold: $X_6\subset {\mathbb {P}}^5$
  8. Quintic fibration over ${\mathbb {P}}^1$: $X_{2,5}\subset{\mathbb {P}}^1\times{\mathbb {P}}^4$
  9. Resolved determinantal sextic in ${\mathbb {P}}^5$: $X_A\subset{\mathbb {P}}^5\times{\mathbb {P}^5}$
  10. Complete intersection in Grassmannian: $X_{1^8}\subset G(2,8)$
  11. Local toric Calabi-Yau varieties
  12. Threefolds
  13. Resolved conifold: ${\cal O}(-1)\oplus{\cal O}(-1)\to {\mathbb {P}}^1$
  14. Local Hirzebruch surface: ${\cal O}(-2,-2)\to {\mathbb {P}}^1\times{\mathbb {P}}^1$
  15. Fourfolds
  16. ...and 8 more sections