The B-model connection and mirror symmetry for Grassmannians

• Published in: Advances in mathematics (New York. 1965) Vol. 366; p. 107027
• Main Authors: Marsh, R.J., Rietsch, K.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 03.06.2020
• Subjects: Cluster algebras; Gauss-Manin system; Grassmannian quantum cohomology; Gromov-Witten theory; Landau-Ginzburg model; Mirror symmetry; Gromov-Witten theory; Gauss-Manin system; Landau-Ginzburg model; Grassmannian quantum cohomology; Mirror symmetry; Cluster algebras
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2020.107027

We consider the Grassmannian X=Grn−k(Cn) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ∘,Wq:Xˇ∘→C), where Xˇ∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian Xˇ, and we express W succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to Wq a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontanine, Kim and van Straten.