Genus one mirror symmetry for intersection of two cubics in $\mathbb{P}^5$

TL;DR

This work proves genus-one BCOV mirror symmetry for the Calabi–Yau threefold $X_{3,3}$, a smooth complete intersection of two cubics in $\mathbb{P}^5$, by leveraging the Batyrev–Borisov mirror and toric desingularizations. The authors combine arithmetic Riemann–Roch to express $\tau_{BCOV}$ up to a holomorphic factor $|\exp(-F_{1,B})|$ and a rational function $|\phi|$, with a detailed period computation and an adapted basis that align $F_{1,B}$ with Popa’s genus-one Gromov–Witten generating function $F_{1,A}$. They compute the genus-one GW invariants $N_1^d(X_{3,3})$ and, through a careful analysis of L^2-norms, Yukawa coupling, and Schmid-type asymptotics, determine the explicit divisor of $|\phi|$ as $|\psi^{-68}(\psi^6-1)^{7/3}|$ up to a constant. The result confirms the genus-one BCOV conjecture for this non-hypersurface, toric-complete intersection, using heavy toric geometry and computer-assisted Euler characteristic calculations. The methods extend BCOV-type genus-one mirror symmetry to a broader class of complete intersections in toric settings and showcase the role of explicit toric mirrors and arithmeticRR in bridging analytic and GW data.

Abstract

This paper establishes BCOV-type genus one mirror symmetry for the intersections of two cubics in $\mathbb{P}^5$. The proof applies previous constructions of the mirror family by the second author and computations of genus one Gromov-Witten invariants by A. Popa. The approach adapts the strategy used for hypersurfaces, as developed by the first author and collaborators, but addresses the distinct geometry involved. A key feature is a systematic usage of toric techniques and related computer aided calculations to determine seemingly otherwise inaccessible invariants.

Theorems & Definitions (41)

• Theorem 1.1
• Proposition 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3
• Proposition 4.1
• Proposition 4.2