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Polynomial Superpotential for Grassmannian \({\text {Gr}}(k,n)\) from a Limit of Vertex Function

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Abstract

In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, \(X=T^{*}{\text {Gr}}(k,n)\). This integral representation can be used to compute the \(\hbar \rightarrow \infty \) limit of the vertex function, where \(\hbar \) denotes the equivariant parameter of a torus acting on X by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the A-series of the Grassmannian \({\text {Gr}}(k,n)\) with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.

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Acknowledgements

We thank Thomas Lam for very useful comments. Work of A. Smirnov is partially supported by NSF grant DMS-2054527 and by the RSF under grant 19-11-00062. Work of A. Varchenko is partially supported by NSF grant DMS-1954266.

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  1. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3250, USA

    Andrey Smirnov & Alexander Varchenko

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  1. Andrey Smirnov
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  2. Alexander Varchenko
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Correspondence to Andrey Smirnov.

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Smirnov, A., Varchenko, A. Polynomial Superpotential for Grassmannian \({\text {Gr}}(k,n)\) from a Limit of Vertex Function. Arnold Math J. (2024). https://doi.org/10.1007/s40598-024-00245-w

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