Calabi–Yau Integrals in Geometry & Physics

• Calabi–Yau integrals are period integrals of holomorphic volume forms over cycles in CY manifolds, crucial for mirror symmetry and enumerative geometry.
• Their computation utilizes Picard–Fuchs differential equations, Frobenius methods, and high-precision numerical techniques to expose deep geometric and arithmetic structures.
• The arithmetic integrality of mirror maps and related invariants links these integrals to p-adic analysis and modular forms, bridging complex geometry with quantum field theory.

Calabi-Yau integrals are period-type integrals associated with Calabi-Yau manifolds and play a foundational role in mirror symmetry, string compactification, the paper of Feynman amplitudes, and arithmetic geometry. In both mathematical and physical settings, such integrals involve the analytic computation of periods of holomorphic top-forms over cycles (or chains) on families of Calabi-Yau (CY) varieties; their structure encodes deep geometric, topological, and arithmetic information.

1. Formalism of Calabi–Yau Integrals

At the core, a Calabi-Yau integral is the period of the nowhere-vanishing holomorphic volume form Ω over a cycle γ in the appropriate (co)homology group, typically

$\int_{\gamma} \Omega,$

where, for a Calabi-Yau n-fold $X$, one has $\Omega \in H^{n,0}(X)$. For local (noncompact) Calabi-Yau geometries, open-string sectors and D-brane boundary conditions naturally lead to integrals over chains with boundary, pairing the holomorphic form with 3-chains ($\Gamma \subset X$) whose boundaries lie on prescribed holomorphic curves. The computation of these periods is governed by the Gauss–Manin connection and, more concretely, by the Picard–Fuchs differential equations characterizing the variation of Hodge structure.

The calculation often proceeds by reducing the multi-dimensional integral, through symmetry and coordinate choices, to lower-dimensional polylogarithmic, elliptic, or more general Calabi–Yau period integrals depending on the complexity and the number of complex structure moduli.

2. Picard–Fuchs Equations and Period Structure

Periods of holomorphic top-forms satisfy linear differential equations with regular singularities—Picard–Fuchs equations—whose structure is dictated by the variation of Hodge structure on the (co)homology of the family. For one-parameter families with $h^{n-1,1} = 1$, the Picard–Fuchs operator is of order $n+1$ (for $n$-folds). Its solutions, in the vicinity of a maximally unipotent monodromy (MUM) point, organize as a Frobenius basis: $$\varpi_0(z),\quad \varpi_1(z) = \varpi_0(z) \log z + \cdots, \quad \ldots$$ In the context of chain integrals arising from open-string configurations, the periods instead satisfy inhomogeneous Picard–Fuchs systems, with inhomogeneities arising from exact forms localized near the boundaries of the chain. Direct construction of these differential operators can employ Griffiths–Dwork reduction or rescaling algorithms specific to the defining equations of the CY geometry (Fuji et al., 2010).

3. Arithmetic Integrality and Mirror Symmetry

In local mirror symmetry, a distinctive phenomenon emerges: the integrality of the expansions of mirror maps, their inverses, and exponential functions appearing in mirror curve equations. The series coefficients of these maps (e.g., open–closed mirror maps)

$$q_i = \exp\left(\frac{g_i}{g_0}\right) \in z_i \cdot \mathbb{Z}[z_1,\ldots,z_N]$$

are shown to be integral under certain combinatorial conditions on the charge vectors defining the geometry (Zhou, 2010). The proof leverages $p$-adic congruence properties of multinomial coefficients and Dwork's Lemma, which gives a $p$-adic criterion for the integrality of power series. Moreover, these integrality properties extend to the inverses of the mirror maps (by Lagrange–Good inversion), and to functions defining mirror curves, such as

$$y = \exp(-W),\qquad \exp(-W) \in \mathbb{Z}[z_0,\ldots,z_N].$$

This arithmetic structure is reminiscent of classical modular objects and is critical for the calculation of enumerative invariants, such as instanton numbers, which are expected to be integral.

4. Numerical and Algorithmic Calculations of Period Integrals

The practical computation of Calabi–Yau integrals, especially periods for compact and rigid CY threefolds, often relies on high-precision numerical methods. For example, for double covers of $\mathbb{P}^3$ branched along arrangements of planes (“double octic” CYs), or for Schoen’s threefolds (fiber products of elliptic surfaces with small resolutions), the period integral can be reduced to explicit triple integrals over real cells: $$\int_{\gamma} \omega = \int_{\gamma} \frac{dx\,dy\,dz}{\sqrt{F(x,y,z)}},$$ with $F$ a product of linear forms. The singular behavior near the boundary of integration is tamed via affine coordinate transformations and substitutions, allowing for stable, precise numerical integration (frequently implemented in Maple). The resultant period vectors have been checked for commensurability with those from associated modular forms, providing evidence for arithmetic correspondences (e.g., to weight-four cusp forms and Kuga–Sato modular varieties) (Cynk et al., 2017, Đonlagić, 13 Apr 2025).

Alternatively, periods can be computed via the corresponding Picard–Fuchs operator and the Frobenius method, with analytic continuation through carefully constructed paths in moduli space (Chmiel, 2019).

5. Calabi–Yau Integrals in Feynman Amplitudes and Quantum Field Theory

Beyond pure geometry, Calabi–Yau integrals appear as the non-polylogarithmic sectors of multi-loop Feynman integrals in quantum field theory. Classic examples are traintrack or fishnet diagrams in $\varphi^4$ theory and beyond, where the multi-loop scalar integrals are recast as period integrals over families of algebraic varieties—K3 surfaces (for three loops), Calabi–Yau threefolds (for four loops), and so on: $y^2 = 4x^3 - g_2(\vec{z})\,x - g_3(\vec{z}),$ with $g_2$, $g_3$ polynomials in auxiliary variables parameterizing the Feynman parametrization (Bourjaily et al., 2018). The rigidity of an integral (the minimal dimension over which the residual non-polylogarithmic piece lives) saturates at $2(L-1)$ for marginal $L$-loop integrals in four dimensions, and the algebraic variety defined by the residual integration is Calabi–Yau precisely when the corresponding Feynman parametric representation satisfies the Calabi–Yau condition (degree equals sum of weights) (Bourjaily et al., 2018).

Canonical differential equations for such Feynman integrals—with or without multiple scales—can often be constructed by aligning the master-integral basis with the underlying (mixed) Hodge structure of the geometry. Transformation to “$\varepsilon$-factorised” form in dimensional regularization enables efficient order-by-order integration (Pögel et al., 2022, Frellesvig et al., 16 Dec 2024, Duhr et al., 26 Mar 2025, Maggio et al., 24 Apr 2025).

6. Role in Enumerative Geometry and Mirror Computations

Calabi–Yau integrals are intimately connected to enumerative invariants in Gromov–Witten theory and mirror symmetry. Period integrals are identified with generating functions for intersection numbers on moduli spaces of stable maps or quasimaps (e.g., $CP^1 \to CP^{N-1}$ with marked points); they also encode the mirror transformation (mirror map) and provide explicit solutions to the Picard–Fuchs equations arising in the computation of Gromov–Witten invariants (Jinzenji et al., 2022). In toric settings, generalized hypergeometric (GKZ) $\mathscr{D}$-modules governing period integrals are shown to be “complete”—all their solutions are realized as periods, making them equivalent to the Picard–Fuchs systems of the family (Lee, 2022).

7. Broader Arithmetic and Motivic Perspective

From the motivic viewpoint, Calabi–Yau integrals are periods of motives associated with the varieties. The relevant cohomology—Betti, de Rham, and $\ell$-adic (étale)—carries structures (Hodge filtration, Galois action) which intersect in the definition of a motive. The periods—numbers appearing in the explicit comparison of these realizations—are central objects of paper in the theory of motives, and conjecturally relate to critical values of $L$-functions via rationality properties summarized in the formalism of Deligne's period conjecture (Bönisch et al., 2021).

The maximal cut of Feynman integrals associated to Calabi–Yau motives thus directly yields periods of the corresponding varieties, with the variation across parameters (e.g., kinematic invariants in physics or moduli in geometry) controlled by the Gauss–Manin connection and leading to rich arithmetic phenomena, including connections to modular forms and algebraic correspondences.

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In summary, Calabi–Yau integrals are central mathematical and physical objects at the interface of Hodge theory, mirror symmetry, enumerative geometry, quantum field theory, and arithmetic geometry; their computation—analytic, numerical, and algebraic—illuminates deep structures underlying periods, special functions, modularity, and the arithmetic of both classical and quantum systems.