Quantum periods of Calabi-Yau fourfolds

TL;DR

Problem: understand quantum periods and their enumerative content for Calabi–Yau fourfolds, focusing on a single Kähler modulus. Approach: derive Picard--Fuchs operators from GLSMs, construct integral quantum periods via brane charges and monodromies, and extract genus-zero GW, Klemm–Pandharipande invariants, and genus-one BPS data using numerical analytic continuation; verify with intersection-theory checks. Findings: large-volume points are regular singular points with non-maximally unipotent monodromy in order-six PF systems, yielding integral, instanton-generated quantum periods and consistent genus-zero and genus-one invariants across explicit examples; monodromy integrality is confirmed. Significance: demonstrates richer middle-cohomology structure in CY4s beyond CY3s, provides a robust computational pipeline for quantum cohomology and GW enumerations, with potential implications for string cosmology and open-closed dualities.

Abstract

In this work we study the quantum periods together with their Picard-Fuchs differential equations of Calabi-Yau fourfolds. In contrast to Calabi-Yau threefolds, we argue that the large volume points of Calabi-Yau fourfolds generically are regular singular points of the Picard-Fuchs operators of non-maximally unipotent monodromy. We demonstrate this property in explicit examples of Calabi-Yau fourfolds with a single Kahler modulus. For these examples we construct integral quantum periods and study their global properties in the quantum Kahler moduli space with the help of numerical analytic continuation techniques. Furthermore, we determine their genus zero Gromov-Witten invariants, their Klemm-Pandharipande meeting invariants, and their genus one BPS invariants. In our computations we emphasize the features attributed to the non-maximally unipotent monodromy property. For instance, it implies the existence of integral quantum periods that at large volume are purely worldsheet instanton generated. To verify our results, we also present intersection theory techniques to enumerate lines with a marked point on complete intersection Calabi-Yau fourfolds in Grassmannian varieties.

Figures (1)

• Figure 2.1: Partwise illustration of a Kähler moduli space, which shows three regular singular points, $z_1$, $z_2$ and $z_3$. The solutions $\Pi^{(k)}_{\alpha}$ for $\alpha=1,2,3$ converge within circles around $z_\alpha$ whose radii are given by the distance to the closest other $z_\alpha$. On the overlaps of convergence areas --- such as the intersecting region of the circles around $z_1$ and $z_2$ --- there is a $\operatorname{GL}(n,\mathbb{C})$ transformation relating the respective solutions $\Pi^{(k)}_{\alpha}$.