Affine monodromy, integrality, and uniqueness of the Loftin–Yau–Zaslow semi-flat Calabi–Yau model at the 'Y' vertex

The SYZ picture of mirror symmetry predicts that near the large complex structure limit a Calabi–Yau threefold carries a special Lagrangian torus fibration over an integral affine base whose singular locus is a trivalent graph, and a local metric model at a trivalent 'Y' vertex was a recognized missing ingredient. Loftin, Yau and Zaslow constructed one: solving the elliptic affine-sphere equation on the thrice-punctured sphere with a cubic differential having three double poles, and applying the Baues–Cortés cone construction, yields a semi-flat Calabi–Yau metric solving the real Monge–Ampère equation on a 3-ball minus a Y-shaped vertex set. Their paper and its 2008 erratum leave the model explicitly incomplete: the equiaffine monodromy representation of the construction was never computed, so it is unknown whether it lies in $\mathrm{SL}(3,\mathbb{Z})\ltimes\mathbb{R}^3$ and the flat fibers descend to genuine special Lagrangian 3-tori; uniqueness of the solution with the prescribed pole asymptotics is unknown (the maximum principle fails); and the model could not be compared with the combinatorial predictions of Gross–Siebert and Haase–Zharkov. Later work sharpened but did not close these questions: Dunajski–Plansangkate identified the integrable structure (Painlevé III for radial reductions), Yang Li constructed competing non-semi-flat Taub-NUT/Ooguri–Vafa type vertex metrics, and Heller–Ouyang–Pedit resolved the existence-with-integrality question in the hyperbolic affine-sphere variant while explicitly leaving the monodromy of the original elliptic construction unknown. The problem is to complete the Loftin–Yau–Zaslow model: compute its monodromy and decide integrality; decide uniqueness of the solution; and determine whether the resulting affine structure matches the combinatorially predicted one.

Background

Mirror symmetry predicts that Calabi–Yau manifolds (compact Kähler manifolds with trivial canonical bundle) come in mirror pairs, and the SYZ conjecture explains the duality geometrically: near the 'large complex structure limit' a Calabi–Yau $n$-fold should carry a fibration by special Lagrangian tori — submanifolds that are Lagrangian for the Kähler form and on which the holomorphic volume form restricts to a fixed-phase volume form — over a base $B$ carrying an integral affine structure, i.e., a flat, torsion-free connection with a parallel volume form whose monodromy takes values in the arithmetic group $\mathrm{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^n$. The simplest compatible metrics are the semi-flat ones, which are flat along the torus fibers; such a metric on the total space of the tangent bundle $TB$ is determined by a Hessian potential $\Phi$ on $B$ (the base metric is $\Phi_{ij}\,dx^i dx^j$ in affine coordinates) satisfying the real Monge–Ampère equation $\det(\Phi_{ij})=1$, which is equivalent to Ricci-flatness. In complex dimension two this picture was established for elliptically fibered K3 surfaces by Gross–Wilson, using an explicit local metric near each singular fiber (the Ooguri–Vafa or 'stringy cosmic string' metric) — and even there the local model alone did not suffice for the global statement. In dimension three the expected singular locus on $B$ is a trivalent graph; local Monge–Ampère models were known near smooth points and near the edges of the graph, but the trivalent vertex — the 'Y' vertex — had none. On the algebro-geometric side, Gross–Siebert (from logarithmic degeneration data) and Haase–Zharkov constructed combinatorial integral affine structures with singularities; however, the affine Kähler metrics Haase–Zharkov build on them do not satisfy the Monge–Ampère equation, so an analytic local model was still missing.

Loftin–Yau–Zaslow supplied it via affine differential geometry. An affine sphere is a hypersurface in $\mathbb{R}^{n+1}$ whose affine normal lines (a canonical transverse line field, defined relative to the equiaffine group $\mathrm{SL}(n+1,\mathbb{R})\ltimes\mathbb{R}^{n+1}$ of volume-preserving affine transformations) all pass through one point — the elliptic type, with center on the concave side — or are parallel (parabolic type); the remaining centered type is hyperbolic. In dimension two the intrinsic (Blaschke) metric of an elliptic affine sphere is conformal, $e^{\psi}|dz|^2$, and together with its Pick form — a holomorphic cubic differential $U\,dz^3$ measuring deviation from a quadric — satisfies the affine-sphere equation $\psi_{z\bar z}+\tfrac12 e^{\psi}+|U|^2 e^{-2\psi}=0$ (an equation of Tzitzeica type, a classical family of second-order nonlinear equations from affine differential geometry). Conversely, a solution on a Riemann surface $\Sigma$ determines an immersion of the universal cover into $\mathbb{R}^3$ (the Simon–Wang developing map), equivariant under an equiaffine monodromy representation $\rho:\pi_1(\Sigma)\to\mathrm{SL}(3,\mathbb{R})$. Baues and Cortés proved that the metric cone over a centered elliptic affine sphere carries a parabolic affine sphere structure, i.e., a convex potential solving $\det(\Phi_{ij})=1$; the 2008 erratum extends the coning to hyperbolic affine spheres. Coning thus converts an elliptic affine sphere on $\Sigma=\mathbb{CP}^1\setminus\{p_1,p_2,p_3\}$ into a Monge–Ampère solution on a 3-ball minus a trivalent 'Y'-shaped set whose three rays correspond to the three punctures, hence into a semi-flat Calabi–Yau metric with flat special Lagrangian fibers. Loftin–Yau–Zaslow proved, by the direct method of the calculus of variations, that the affine-sphere equation admits a solution on the thrice-punctured sphere with prescribed pole asymptotics (their radially symmetric local model near each pole) when $U$ has double poles at the three punctures — the singularity type dictated by the SYZ picture. The 2008 erratum corrects a gap in the original existence proof: Theorem 2′ establishes existence only for cubic differentials of sufficiently small norm (no explicit bound), and records an alternative route through hyperbolic affine spheres, for which only the holonomy type around each puncture was known.

Three issues were then explicitly open. First, the flat fibers of the coned metric are special Lagrangian 3-planes but descend to 3-tori only if the affine monodromy lies in $\mathrm{SL}(3,\mathbb{Z})\ltimes\mathbb{R}^3$; the monodromy representation — obtainable in principle by integrating the affine-sphere structure equations, a first-order system in the frame $\{f,f_z,f_{\bar z}\}$, around loops generating $\pi_1(\Sigma)$ — was not computed, even though the local model near each pole was understood. Second, uniqueness of the constructed solution within the prescribed singularity class was unknown, the maximum principle being unavailable for an equation carrying both $e^{-2\psi}$ and $e^{\psi}$ nonlinearities. Third, without the monodromy one could not verify that the resulting affine structure is the one predicted by the combinatorial constructions of Gross–Siebert and Haase–Zharkov, leaving open whether this local analytic model is in fact the model of actual Calabi–Yau degenerations near a trivalent vertex.

Problem Statement

Complete the local model of the trivalent ('Y'-vertex) singularity of the SYZ base constructed by Loftin–Yau–Zaslow, by settling the three questions their paper (J. Differential Geom. 71 (2005) 129–158) and its 2008 erratum explicitly leave open, for the object as corrected by the erratum.

Setting. $\Sigma=\mathbb{CP}^1\setminus\{p_1,p_2,p_3\}$ with a holomorphic cubic differential $U\,dz^3$ having double poles at the three punctures (up to the rigidity of the pole triple, a one-complex-parameter family given by the scale of $U$); the elliptic affine-sphere metric $e^{\psi}|dz|^2$ solving $\psi_{z\bar z}+\tfrac12 e^{\psi}+|U|^2 e^{-2\psi}=0$ with the prescribed pole asymptotics (the radially symmetric local model near each pole), known to exist for sufficiently small $U$ by the erratum's Theorem 2′; the associated developing map with equiaffine monodromy $\rho:\pi_1(\Sigma)\to\mathrm{SL}(3,\mathbb{R})$; and the semi-flat Calabi–Yau metric on the cone $B$ (a 3-ball minus a Y-shaped vertex set) obtained by the Baues–Cortés coning.

(1) Monodromy and integrality. Compute the monodromy representation $\rho$ of the Loftin–Yau–Zaslow elliptic affine sphere — equivalently, of the flat special affine structure on $B$ — and determine whether $\rho$, together with its translational (period) data, is conjugate into $\mathrm{SL}(3,\mathbb{Z})\ltimes\mathbb{R}^3$ — determined as a function of the scaling parameter of $U$ over the small-norm existence regime. Equivalently: decide whether the fibers of the semi-flat Calabi–Yau metric descend to genuine special Lagrangian 3-tori.

(2) Uniqueness. Decide whether the solution of the elliptic affine-sphere equation with the prescribed singularity class — in the small-norm regime where existence is known — is unique. The maximum principle is unavailable because the equation carries both $e^{-2\psi}$ and $e^{\psi}$ nonlinearities; the two possible outcomes are a uniqueness theorem in the weak $W^{1,2}_{\mathrm{loc}}(\Sigma)$ class of solutions with the prescribed pole asymptotics, or an explicit (analytic or certified numerical) construction of two distinct solutions with identical singularity data.

(3) Model consistency. Using (1), verify or refute that the integral affine structure and Monge–Ampère metric produced by the Loftin–Yau–Zaslow construction coincide with the local structure at a trivalent vertex predicted by the combinatorial/algebro-geometric models of Gross–Siebert and Haase–Zharkov.

The parts are linked: (2) settles whether 'the' monodromy in (1) is canonical; (1) is precisely the input that unlocks the comparison (3), which gives the model its mirror-symmetric meaning. The broader programmatic question — gluing such local models into global Calabi–Yau degenerations, including instanton corrections — is the motivating context and lies outside the scope of the three questions.

References

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