Trenner and Wilson conjectured (Conjecture 3.8 of their 2011 Journal of Geometric Analysis paper; Conjecture 2.8 in the arXiv version) that the Kähler metric with potential −log⁡f-\log f on the complexified index cone of a real cubic form ff — the asymptotic mirror Weil–Petersson (AMWP) metric that models the Weil–Petersson metric of a mirror Calabi–Yau threefold near a large complex structure limit — has curvature tensor g⊙gg\odot g minus a correction term built from the cubic's constant third partial derivatives, divided by 64f264f^2. They proved this identity only for ternary cubics (r=3r=3) and expected the general case to follow from projective special Kähler geometry. In the current literature, a claimed general proof exists only in an unpublished, essentially uncited 2010 arXiv preprint by Trenner himself, while no refereed proof or counterexample has appeared; and the asymptotic mirror symmetry identification underlying the AMWP model remains established only under toric hypotheses. The problem asks for a rigorously verified proof or explicit counterexample to the conjectured curvature identity for arbitrary cubics, which would make the AMWP curvature calculus — including the counterexamples to Weil–Petersson scalar-curvature non-positivity derived from it — available at arbitrary rank rather than only for ternary cubics.

Background

A Calabi–Yau threefold VV is a smooth complex projective threefold with trivial canonical bundle and H1(OV)=0H^1(\mathcal{O}_V)=0. Its Kähler classes form the Kähler cone inside H2(V,R)H^2(V,\mathbb{R}); after complexification one gets coordinates ti=xi+iyit_i=x_i+iy_i on H2(V,C)H^2(V,\mathbb{C}), on which the triple cup product defines a cubic form f(y)=∫Vy3f(y)=\int_V y^3, the cubic intersection form. Mirror symmetry predicts a mirror partner V∘V^\circ whose complex-structure deformations correspond to the complexified Kähler deformations of VV, the correspondence concentrating near a large complex structure limit point of V∘V^\circ — a boundary point of its complex-structure moduli space at which the monodromy acting on periods of the holomorphic 3-form is maximally unipotent, mirrored by the large radius limit of VV. The complex-structure moduli space carries the Weil–Petersson metric, the natural L2L^2 metric on periods of the holomorphic 3-form. For Calabi–Yau manifolds this metric is special Kähler: special Kähler geometry equips a Kähler manifold with a flat, torsion-free connection compatible with the complex structure and the Kähler form, and quotienting a conic affine special Kähler manifold by its natural C∗\mathbb{C}^*-action yields a projective special Kähler manifold, the structure carried by Calabi–Yau moduli (formalized by Freed following the physics of N=2N=2 vector multiplets). Strominger proved that the Weil–Petersson curvature tensor equals a term of the form g⊙gg\odot g (the curvature pattern of complex space forms) minus a correction term built from the Yukawa couplings — the symmetric cubic of third derivatives of the holomorphic prepotential, which on the mirror side encodes the intersection form.

Trenner and Wilson studied the metric on the Kähler side that models the Weil–Petersson metric near the boundary while discarding instanton corrections. On the index cone of the cubic ff — the open cone where f>0f>0 and the Hessian of ff has signature (1,r−1)(1,r-1), which contains the Kähler cone by the Hodge index theorem — the potential −log⁡f(y1,…,yr)-\log f(y_1,\dots,y_r) defines a Kähler metric, the asymptotic mirror Weil–Petersson (AMWP) metric. The underlying asymptotic mirror symmetry correspondence asserts that the Gauss–Manin prepotential of the mirror agrees with f/6f/6 up to lower-order terms near a maximally unipotent boundary point; Trenner and Wilson verified this in detail for toric hypersurface mirrors, using mirror theorems and generalized hypergeometric (GKZ) systems for the periods. With the AMWP metric they computed boundary curvature behavior — for instance Ricci curvature tending to −∞-\infty — and constructed counterexamples to a conjecture of Ooguri and Vafa, motivated by string-theoretic swampland considerations, that the Weil–Petersson scalar curvature should stay non-positive in a neighborhood of the large complex structure limit point.

These curvature computations rest on an algebraic identity that the paper proves only for ternary cubics (r=3r=3), by reduction to Weierstrass canonical form, and conjectures for arbitrary cubics: the curvature tensor of the AMWP metric should be given by the Strominger-type formula with the correction term built from the cubic's constant third partial derivatives. The authors expected the general case to follow from the theory of projective special Kähler manifolds. A related point of context concerns the logical status of the mirror-symmetry input to the model: the asymptotic identification of prepotentials is established only under toric hypotheses, and how far it holds beyond them is unknown. The curvature conjecture itself, by contrast, is framed in the paper as a self-contained algebraic identity that should hold for arbitrary cubics on their complexified index cones, independently of the mirror geometry.

Problem Statement

Let f(y1,…,yr)f(y_1,\dots,y_r) be a cubic form with real coefficients in rr variables, and let W⊂RrW\subset\mathbb{R}^r be its index cone, the open cone on which f>0f>0 and the Hessian (∂2f/∂yi∂yj)(\partial^2 f/\partial y_i\partial y_j) has signature (1,r−1)(1,r-1). On the complexified cone with coordinates ti=xi+iyit_i=x_i+iy_i and y∈Wy\in W, the Kähler potential −log⁡f(y1,…,yr)-\log f(y_1,\dots,y_r) defines the AMWP metric gijˉ=∂2(−log⁡f)/∂ti∂tˉjg_{i\bar{j}}=\partial^2(-\log f)/\partial t_i\partial\bar{t}_j.

Prove, or disprove by explicit counterexample, that for every rr and every real cubic ff the curvature tensor of this metric on the complexified index cone satisfies

Rijˉklˉ=gijˉgklˉ+gilˉgkjˉ−∑p,qgpqˉfikpfjlq64f2,R_{i\bar{j}k\bar{l}}=g_{i\bar{j}}g_{k\bar{l}}+g_{i\bar{l}}g_{k\bar{j}}-\sum_{p,q}\frac{g^{p\bar{q}}f_{ikp}f_{jlq}}{64f^{2}},

where fikp=∂3f/∂yi∂yk∂ypf_{ikp}=\partial^3 f/\partial y_i\partial y_k\partial y_p denote the constant third partial derivatives of ff and (gpqˉ)(g^{p\bar{q}}) is the inverse matrix of (gijˉ)(g_{i\bar{j}}). Trenner and Wilson established this identity only for ternary cubics (r=3r=3) and conjectured it for arbitrary cubics, expecting a proof via the theory of projective special Kähler manifolds.

References

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