Main Conjecture: Genus-1 GW equals elliptic virtual structure B-model

Establish that for any projective hypersurface M_N^k with N ≥ k, the genus-1 A-model generating function F_{1}^{N,k,A}(t^0,t^1,…,t^{N−2}), defined via Gromov–Witten invariants, equals the B-model generating function F_{1,vir.}^{N,k,B}(x^0(t^*),x^1(t^*),…,x^{N−2}(t^*)) computed from elliptic multi-point virtual structure constants, where x^p(t^*) is the inverse of the mirror map built from genus-0 virtual structure constants.

Background

After defining elliptic multi-point virtual structure constants and their generating function F_{1,vir.}^{N,k,B}, the authors state their central conjecture: the equality of the genus-1 Gromov–Witten generating function (A-model) and the B-model generating function obtained from virtual structure constants via the inverse mirror map.

This conjecture generalizes the mirror symmetry approach to genus-1 for both Calabi–Yau and Fano projective hypersurfaces and is supported by numerical tests presented later in the paper. A formal proof is not provided.