in the celebrated work of bershadsky--cecotti--ooguri--vafa the genus one string partition function in the b-model is identified with certain analytic torsion of the hodge laplacian on a k$\ddot{\text{a}}$hler manifold. in a joint work with shu shen (imj-prg) and jianqing yu (ustc) we study the analogous torsion in landau--ginzburg models. i will explain the corresponding index theorem based on the asymptotic expansion of the heat kernel of the schr$\ddot{\text{o}}$dinger operator. i will also explain the rigorous definition of the bcov torsion for homogeneous polynomials on ${\mathbb c}^n$. lastly i will explain the conjecture stating that in the calabi--yau case the bcov torsion solves the holomorphic anomaly equation for marginal deformations.