Research output: Contribution to journal › Article

**The B-model connection and mirror symmetry for Grassmannians.** / Marsh, Robert; Rietsch, Konstanze.

Research output: Contribution to journal › Article

Marsh, R & Rietsch, K 2020, 'The B-model connection and mirror symmetry for Grassmannians', *ADVANCES IN MATHEMATICS*, vol. 366, 107027. https://doi.org/10.1016/j.aim.2020.107027

Marsh, R., & Rietsch, K. (2020). The B-model connection and mirror symmetry for Grassmannians. *ADVANCES IN MATHEMATICS*, *366*, [107027]. https://doi.org/10.1016/j.aim.2020.107027

Marsh R, Rietsch K. The B-model connection and mirror symmetry for Grassmannians. ADVANCES IN MATHEMATICS. 2020 Jun 3;366. 107027. https://doi.org/10.1016/j.aim.2020.107027

@article{c97402141c7d4304a6ad796f0591417e,

title = "The B-model connection and mirror symmetry for Grassmannians",

abstract = "We consider the Grassmannian X=Gr n−k(C n) and describe a {\textquoteleft}mirror dual{\textquoteright} 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 Pl{\"u}cker 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. ",

keywords = "Cluster algebras, Gauss-Manin system, Grassmannian quantum cohomology, Gromov-Witten theory, Landau-Ginzburg model, Mirror symmetry",

author = "Robert Marsh and Konstanze Rietsch",

year = "2020",

month = jun,

day = "3",

doi = "10.1016/j.aim.2020.107027",

language = "English",

volume = "366",

journal = "ADVANCES IN MATHEMATICS",

issn = "0001-8708",

publisher = "ACADEMIC PRESS INC",

}

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

VL - 366

JO - ADVANCES IN MATHEMATICS

JF - ADVANCES IN MATHEMATICS

SN - 0001-8708

M1 - 107027

ER -

King's College London - Homepage

© 2020 King's College London | Strand | London WC2R 2LS | England | United Kingdom | Tel +44 (0)20 7836 5454