Equality of elliptic virtual structure constants with the BCOV–Zinger B-model for Calabi–Yau hypersurfaces

Establish that for a Calabi–Yau hypersurface M_k^k, the B-model generating function of elliptic virtual structure constants F_{1,vir.}^{k,k,B}(x) equals the BCOV–Zinger B-model function F_{1}^{k,k,B}(x), which is expressed in terms of analytic torsion and the power series \tilde{L}_{p}^{k,k}(e^{x}).

Background

In the Calabi–Yau case (N=k), the structure of the generating functions simplifies, and the mirror map t(x) coincides with the one used in the BCOV–Zinger framework. The authors formulate a conjecture identifying their B-model function based on elliptic virtual structure constants with the BCOV–Zinger B-model function.

They provide a reformulation of this conjecture in terms of identities involving loop graph residue contributions and logarithms of \tilde{L}_{p}^{k,k}(e^{x}), and they report numerical confirmation up to degree d=5 for k=4–8, but no complete proof is given.