The B-model connection and mirror symmetry for Grassmannians

TY - JOUR

T1 - The B-model connection and mirror symmetry for Grassmannians

AU - Marsh, Robert

AU - Rietsch, Konstanze

PY - 2020/6/3

Y1 - 2020/6/3

N2 - We consider the Grassmannian X=Gr n−k(C n) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ ∘,W q: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 W q 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.

AB - We consider the Grassmannian X=Gr n−k(C n) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ ∘,W q: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 W q 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.

KW - Cluster algebras

KW - Gauss-Manin system

KW - Grassmannian quantum cohomology

KW - Gromov-Witten theory

KW - Landau-Ginzburg model

KW - Mirror symmetry

UR - http://www.scopus.com/inward/record.url?scp=85081032160&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2020.107027

DO - 10.1016/j.aim.2020.107027

M3 - Article

SN - 0001-8708

VL - 366

SP - 1

EP - 131

JO - ADVANCES IN MATHEMATICS

JF - ADVANCES IN MATHEMATICS

M1 - 107027

ER -