A Peterson program for general Schubert varieties and mirror symmetry

Abstract

We initiate a `Peterson program' that seeks to extend Dale Peterson's theory for the quantum cohomology of flag varieties to arbitrary Schubert varieties, and that incorporates a mirror-symmetric approach. In this first paper we introduce a generalisation of the (localised) Peterson variety associated to an arbitrary Schubert variety check X_w,P inside a partial flag variety check G/check P. This generalisation is possibly a non-reduced affine scheme that lies in the Langlands dual full flag variety G/B. Additionally, we construct a Lie-theoretic `superpotential' associated to check X_w,P, generalising earlier ones for the flag varieties check G/check P from [Rie08], and we show that its relative critical point locus recovers the generalised Peterson variety. We make the key conjecture that for smooth Fano Schubert varieties the coordinate ring of the generalised Peterson variety recovers the quantum cohomology ring of check X_w,P localised at the quantum parameters. For smooth Fano Schubert varieties X_w,B in check G/check B we furthermore map our generalised Peterson variety to the cotangent bundle 𝒯^*(T) of the maximal torus T of G and construct a partial compactification that we conjecture models the full quantum cohomology ring of check X_w,B, in analogy with the Givental-Kim presentation of QH^*(check G/check B) via the degenerate leaf of the Kostant Toda lattice of check G. These conjectures are verified for all smooth Schubert divisors in the complete type A flag variety, and we show that our superpotential agrees with that introduced for Grassmannian Schubert varieties by Rietsch and Williams, after a suitable isomorphism of the domains.

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BibTeX RIS

BibTeX

@misc{li2026peterson,
  title = {{A Peterson program for general Schubert varieties and mirror symmetry}},
  author = {Changzheng Li and Konstanze Rietsch and Mingzhi Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30242},
  eprint = {2609.30242},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30242},
}

RIS

TY  - UNPB
TI  - A Peterson program for general Schubert varieties and mirror symmetry
AU  - Changzheng Li
AU  - Konstanze Rietsch
AU  - Mingzhi Yang
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30242
UR  - https://arxiv.org/abs/2609.30242
AB  - We initiate a `Peterson program' that seeks to extend Dale Peterson's theory for the quantum cohomology of flag varieties to arbitrary Schubert varieties, and that incorporates a mirror-symmetric approach. In this first paper we introduce a generalisation of the (localised) Peterson variety associated to an arbitrary Schubert variety check X_w,P inside a partial flag variety check G/check P. This generalisation is possibly a non-reduced affine scheme that lies in the Langlands dual full flag variety G/B. Additionally, we construct a Lie-theoretic `superpotential' associated to check X_w,P, generalising earlier ones for the flag varieties check G/check P from [Rie08], and we show that its relative critical point locus recovers the generalised Peterson variety. We make the key conjecture that for smooth Fano Schubert varieties the coordinate ring of the generalised Peterson variety recovers the quantum cohomology ring of check X_w,P localised at the quantum parameters. For smooth Fano Schubert varieties X_w,B in check G/check B we furthermore map our generalised Peterson variety to the cotangent bundle 𝒯^*(T) of the maximal torus T of G and construct a partial compactification that we conjecture models the full quantum cohomology ring of check X_w,B, in analogy with the Givental-Kim presentation of QH^*(check G/check B) via the degenerate leaf of the Kostant Toda lattice of check G. These conjectures are verified for all smooth Schubert divisors in the complete type A flag variety, and we show that our superpotential agrees with that introduced for Grassmannian Schubert varieties by Rietsch and Williams, after a suitable isomorphism of the domains.
ER  -

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