Scope and extensions of the Calabi–Yau-to-curve correspondence for Feynman integrals: multi-parameter families and the full four-loop banana integral

The maximal cut of the equal-mass four-loop banana Feynman integral, long known to be a period of a family of Calabi–Yau threefolds (the Hulek–Verrill family), was shown by Jockers, Kotlewski, Kuusela, McLeod, Pögel, Sarve, Wang and Weinzierl to also be a period of an explicit genus-two curve: its polarised holomorphic Jacobian — a new kind of intermediate Jacobian, defined locally by holomorphic continuation from a real Lagrangian sublocus of the complex-structure moduli space — is a principally polarised Abelian surface and hence the Jacobian of a genus-two curve. The authors posed an extension programme: construct an explicit two-parameter example realising the expected genus-three curve correspondence; determine the exceptional slices on which Calabi–Yau families with more than two parameters meet the Schottky locus of curves; express the full (un-cut) four-loop banana integral in $D=2$ in terms of iterated integrals on the genus-two curve; and extend the correspondence to intermediate Jacobians of higher odd-dimensional Calabi–Yau manifolds. The problem is open.

Background

In perturbative quantum field theory, amplitudes are expanded in a coupling and each term is a sum of Feynman integrals indexed by graphs. In dimensional regularisation one computes in $D=2-2\varepsilon$ space-time dimensions, and each integral family reduces to finitely many master integrals obeying differential equations in the kinematic invariants; if the system can be brought to an ε-factorised form (every matrix entry carrying an explicit power of $\varepsilon$), the masters are solved systematically by iterated integrals whose kernels are built from periods — integrals of algebraic differential forms over cycles — of an auxiliary geometry. Which geometry appears is governed by the graph: for the banana graph with $l$ loops ($l+1$ propagators in parallel between two vertices) with equal masses $M$ and external momentum $p$, the single invariant is $z=M^2/p^2$, and the maximal cut (the residue obtained by putting all propagators on shell) is a period of a Calabi–Yau $(l-1)$-fold, a compact Kähler manifold with trivial canonical bundle. The two-loop banana (sunrise) involves an elliptic curve and evaluates to iterated integrals of modular forms; the four-loop equal-mass banana involves the Hulek–Verrill family of Calabi–Yau threefolds, whose maximal-cut period $\psi_0(z)$ is annihilated by a fourth-order Picard–Fuchs operator (the linear differential operator annihilating the period) with a point of maximal unipotent monodromy (a MUM point, i.e. a degeneration with maximally unipotent monodromy) at $z=0$ and nearest singular point at $z=1/25$.

Jockers et al. (2025) showed that $\psi_0(z)$ is also a period of a genus-two curve. The bridge is the intermediate Jacobian, a complex torus built from the middle cohomology $H^3$ of the threefold: the classical Griffiths intermediate Jacobian varies holomorphically with $z$ but is not an Abelian variety (it carries no positive principal polarisation), while the Weil intermediate Jacobian is a principally polarised Abelian variety — a point of the Siegel upper half-space $\mathbb{H}_2$, the moduli space of principally polarised Abelian varieties of dimension two — but varies non-holomorphically with $z$, since its definition involves complex conjugation. The authors therefore introduced a third object, the polarised holomorphic Jacobian $J_2^{\Delta_{\mathbb{R}}}$: restricted to a real Lagrangian sublocus $\Delta_{\mathbb{R}}$ of the complex-structure moduli space — a real slice such as the physical range $z\in]0,1/25[$ on which the Griffiths and Weil data agree — the two notions coincide, and holomorphic continuation to a tubular neighbourhood $U$ of $\Delta_{\mathbb{R}}$ produces a holomorphic family of principally polarised Abelian surfaces. Because every indecomposable principally polarised Abelian surface is the Jacobian of a genus-two curve (the Schottky problem — deciding which principally polarised Abelian varieties arise as Jacobians of curves — is trivial in dimension two), this yields an explicit curve $C_z$, given in Rosenhain form $v^2=u(u-e_2(z))(u-e_3(z))(u-e_4(z))(u-1)$, a linear combination of whose two holomorphic differentials has all periods annihilated by the same Picard–Fuchs operator as the Calabi–Yau period. The construction is local: it depends on the choice of $\Delta_{\mathbb{R}}$ (a different kinematic range requires a different Lagrangian sublocus, reached by analytically continuing the Calabi–Yau periods), and the branch points $e_i(z)$ are transcendental functions of $z$ (an Igusa invariant, a standard set of absolute invariants of genus-two curves, is provably not algebraic in $z$).

Counting dimensions shows when such a correspondence can exist at all. For a Calabi–Yau threefold family with $m\ge1$ kinematic parameters, the polarised holomorphic Jacobians form an $m$-parameter family of principally polarised Abelian varieties, each of complex dimension $m+1$, describing a locus of codimension $\frac{1}{2}(m^2+m+2)$ in the moduli space $\mathcal{A}_{m+1}$, while the Jacobian locus of genus-$(m+1)$ curves has codimension $\frac{1}{2}(m-1)(m-2)$; assuming the two intersect transversely, the Abelian varieties come from curves only along a slice of dimension $\frac{1}{2}m(5-m)-1$ in the $m$-dimensional parameter space, which is empty unless $1\le m\le4$. For $m=1$ (genus two) and $m=2$ (genus three, where again every indecomposable principally polarised Abelian variety is a Jacobian) the correspondence is expected generically; for $m\ge3$ it requires exceptional slices. The equal-mass four-loop banana ($m=1$) is so far the only worked example.

Problem Statement

The problem is to determine the scope of the Calabi–Yau-to-curve correspondence for Feynman integrals — for which geometries, and for the full integral or only its maximal cut — by resolving the extension programme left open by the construction of Jockers et al. The paper's Conclusions pose the following open directions:

(a) A two-parameter example. For a Feynman integral whose associated Calabi–Yau threefold family depends on two kinematic variables ($m=2$), construct explicitly a two-parameter family of genus-three curves realising the correspondence (matching polarised holomorphic Jacobians), and verify the expectation that the correspondence holds generically, consistent with every indecomposable principally polarised Abelian threefold being the Jacobian of a genus-three curve.

(b) The full banana integral. Determine to what extent the full (un-cut) equal-mass four-loop banana integral in $D=2$ — not only the maximal cut — can be expressed in terms of iterated integrals whose integration kernels live on the constructed genus-two curve; the minimum target is such a representation, and exhibiting it through an ε-factorised system of differential equations is a strengthening. Any such result must be consistent with the known results obtained from Calabi–Yau-period methods.

(c) Exceptional slices for $m>2$. For Calabi–Yau threefold families with more than two kinematic parameters, determine the structure and dimension of the slices in parameter space on which the polarised holomorphic Abelian varieties (of complex dimension $m+1$) land in the Schottky locus $\mathcal{S}_{m+1}$ — the transversality-assumed count predicts dimension $\frac{1}{2}m(5-m)-1$, vanishing outside $1\le m\le4$ — decide whether the transversality assumption holds in concrete families, and construct explicit examples of such correspondences when they exist. The dimension count alone does not settle this direction: it also requires the transversality assumption to be decided in at least one explicit family with $m>2$, together with an explicit example of the resulting curve correspondence whenever the slices are nonempty.

(d) Higher Calabi–Yau dimension. Generalise the construction to intermediate Jacobians of Calabi–Yau manifolds of odd complex dimension greater than three (even-dimensional Calabi–Yau manifolds are excluded, having no canonical symplectic structure on middle cohomology) and clarify when the resulting families of Abelian varieties come from curves. This direction is settled by an explicit instance: for at least one explicit Calabi–Yau family of this kind, extend the construction, decide whether the associated polarised holomorphic Jacobians meet the Schottky locus, and exhibit the resulting curve correspondence where they do — or prove that none arises. A general classification over all such families and dimensions is not required.

These four directions are independent open problems, each a self-contained contribution to the programme that does not presuppose the others; the extension programme as a whole is resolved only once all four are settled.

Each direction is meaningful precisely because the equal-mass four-loop banana is currently the only worked example of the correspondence.

References

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