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Modular Amplitudes and Flux-Superpotentials on elliptic Calabi-Yau fourfolds

Cesar Fierro Cota, Albrecht Klemm, Thorsten Schimannek

TL;DR

The work develops a comprehensive framework for modular amplitudes and flux superpotentials on elliptic Calabi-Yau fourfolds by fixing integral periods through the Gamma class and exploiting Bridgeland/Fourier-Mukai auto-equivalences to realize PSL(2,Z) on the fiber. It shows that genus-zero and genus-one amplitudes are quasi-modular forms obeying holomorphic anomaly equations, and derives explicit modular anomaly equations for π-vertical 4-cycles and 4-point base couplings, illustrated in the X_{24} geometry. The paper combines toric GKZ methods, integral period bases, and brane charge maps to analyze horizontal flux vacua, performing detailed continuations to conifold and orbifold loci and demonstrating flux-aligned stabilization at the conifold but no stable vacua at the orbifold. The results provide a practical route to flux quantization and moduli stabilization in F-theory compactifications and point to extensions to singular fibrations and richer auto-equivalence structures in future work.

Abstract

We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order Kähler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.

Modular Amplitudes and Flux-Superpotentials on elliptic Calabi-Yau fourfolds

TL;DR

The work develops a comprehensive framework for modular amplitudes and flux superpotentials on elliptic Calabi-Yau fourfolds by fixing integral periods through the Gamma class and exploiting Bridgeland/Fourier-Mukai auto-equivalences to realize PSL(2,Z) on the fiber. It shows that genus-zero and genus-one amplitudes are quasi-modular forms obeying holomorphic anomaly equations, and derives explicit modular anomaly equations for π-vertical 4-cycles and 4-point base couplings, illustrated in the X_{24} geometry. The paper combines toric GKZ methods, integral period bases, and brane charge maps to analyze horizontal flux vacua, performing detailed continuations to conifold and orbifold loci and demonstrating flux-aligned stabilization at the conifold but no stable vacua at the orbifold. The results provide a practical route to flux quantization and moduli stabilization in F-theory compactifications and point to extensions to singular fibrations and richer auto-equivalence structures in future work.

Abstract

We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order Kähler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.

Paper Structure

This paper contains 32 sections, 186 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Mathematical and physical structures on the moduli space
  3. The period geometry of Calabi-Yau fourfolds
  4. The structure of $H^{4}(W,\mathbb{Z})$
  5. Fixing an integral basis
  6. B-branes and the asymptotic behaviour of the central charge
  7. Elliptically fibered Calabi-Yau fourfolds
  8. Geometry of non-singular elliptic Calabi-Yau fourfolds
  9. Fourier-Mukai transforms and the ${\rm SL}(2,\mathbb{Z})$ monodromy
  10. Toric construction of mirror pairs
  11. Toric geometry of elliptic fibrations
  12. Picard-Fuchs operators
  13. Amplitudes, geometric invariants and modular forms
  14. Review of genus zero invariants
  15. Genus one invariants
  16. ...and 17 more sections

Figures (3)

  • Figure 1: Schematic structure of the resolved complex structure moduli space of $X_{24}^*$. The large complex structure divisors are shown in blue and the conifold components are red. Exceptional divisors resolving non-normal crossing intersections are indicated with dashed lines.
  • Figure 2: The scalar potential generated by aligned flux, depending on the distance to the conifold $C_1$ in flat coordinates $t_c^1=x+I y,\, t_c^2=0$.
  • Figure 3: The scalar potential generated by a generic choice of flux, depending on the distance to the orbifold $O_1$ in coordinates $o_1=x+I y,\, o_2=0$.