2020/11/19 by Lee, Younggi, Park, Jeehoon, Yim, Jaehyun
#14F25 #14J32 #14J33 #14J81 #53D45 #81T70 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2011.09628
The goal of this article is to provide an explicit algorithmic construction of formal F-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology ℍ of a smooth projective Calabi-Yau complete intersection variety X defined by homogeneous polynomials G1(\underline x), …, Gk(\underline x). Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra A obtained from the twisted de Rham complex which computes ℍ. More explicitly, we introduce a notion of a weak primitive form associated to a solution of the Maurer-Cartan equation of A and the Gauss-Manin connection, which is a weakened version of Saito's primitive form (\citeSaito). In addition, we provide an explicit algorithm for a weak primitive form based on the Gröbner basis in order to achieve our goal. Our approach through the weak primitive form can be viewed as a unifying link (based on Witten's gauged linear sigma model, \citeW93) between the Barannikov-Kontsevich's approach to Frobenius manifolds via dGBV algebras (non-linear topological sigma model, \citeBK) and Saito's approach to Frobenius manifolds via primitive forms and higher residue pairings (Landau-Ginzburg model, \citeST).