Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus

Changzheng Li, Konstanze Rietsch, Mingzhi Yang, Chi Zhang

Abstract

We construct a Plücker coordinate superpotential $\mathcal{F}_-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n_\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class $c_1(\mathbb{ F}\ell(n_\bullet))$ on quantum cohomology are equal to the critical values of $\mathcal{F}_-$. We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for $\mathcal{F}_-$ and the mirror symmetry conjecture.

A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus

Abstract

We construct a Plücker coordinate superpotential that is mirror to a partial flag variety . Its Jacobi ring recovers the small quantum cohomology of and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class on quantum cohomology are equal to the critical values of . We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for and the mirror symmetry conjecture.
Paper Structure (22 sections, 42 theorems, 172 equations)

This paper contains 22 sections, 42 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Langlands dual flag varieties
  5. Open Richardson varieties
  6. Schubert varieties
  7. The Plücker coordinate superpotentials
  8. The superpotential $\mathcal{F}_+$ generalising MaRi
  9. Superpotential $\mathcal{F}_{\operatorname {Lie}}$
  10. The superpotential $\mathcal{F}_-$
  11. Notations for $\mathcal{F}_-$
  12. The description of $\mathcal{Z}$
  13. A Plücker coordinate formula for $\mathcal{F}_-$
  14. Quantum cohomology of partial flag varieties
  15. Peterson isomorphism
  16. ...and 7 more sections

Key Result

Theorem 1.1

There is an isomorphism constructed in e:psi-. The superpotential $\mathcal{F}_-:=\mathcal{F}_{\operatorname{Lie}}\circ \psi_-^{-1}:(P\backslash G)^\circ\times \prod\limits_{i\in I^P} \mathbb{C}^*_q \to \mathbb C$ consists of $(n-1+r)$ summands, The summands satisfy

Theorems & Definitions (102)

  • Theorem 1.1
  • Example 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.10
  • Definition 3.1
  • ...and 92 more