The Genus-One B-Model Calabi-Yau/Landau-Ginzburg Correspondence

Abstract

Let $f:\mathbb C^n\to\mathbb C$ be a homogeneous polynomial of degree $n$ with an isolated critical point at the origin. The Calabi--Yau/Landau--Ginzburg correspondence predicts a close relation between invariants of the Landau--Ginzburg orbifold $(\mathbb C^n/\mathbb Z_n,f)$ and those of the associated Calabi--Yau hypersurface $ X_f=\{[z_1:\cdots:z_n]\in\mathbb P^{n-1}:f(z_1,\ldots,z_n)=0\}. $ In this paper, we establish this correspondence for the genus-one B-model by constructing the Landau--Ginzburg genus-one B-model invariant intrinsically from the $f$-twisted Laplacian on $\mathbb{C}^n/\mathbb{Z}_n$ and proving that it agrees with the corresponding BCOV invariant of $X_f$ up to an additive constant depending only on the dimension.