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Unfolding of equivariant F-bundles and application to the mirror symmetry of flag varieties

Thorgal Hinault, Changzheng Li, Tony Yue YU, Chi Zhang, Shaowu Zhang

TL;DR

This work introduces an equivariant unfolding framework for F-bundles, extending the Hertling-Manin universal unfolding to handle torus actions and infinite-rank settings. Central to the approach are the (IC), (GC), and (GC') conditions that guarantee existence and uniqueness of maximal unfoldings, together with framings that enable control of the u-direction. The authors apply this theory to flag varieties, constructing equivariant big and small quantum D-modules for G/P and proving big quantum cohomology mirror symmetry by unfolding the B-model Landau–Ginzburg data and relating it to the A-model via equivariant F-bundles. The results yield a precise isomorphism of big equivariant D-modules (and hence big quantum cohomology) for flag varieties, with a non-equivariant limit recovering the classical big-mirror statement. This provides a robust, reconstruction-based route to big-quantum mirror symmetry beyond the semisimple or $H^2$-generated cases, and clarifies the role of equivariance in enabling universal unfoldings in Gromov–Witten theory.

Abstract

We establish an unfolding theorem for equivariant F-bundles (a variant of Frobenius manifolds), generalizing Hertling-Manin's universal unfolding of meromorphic connections. As an application, we obtain the mirror symmetry theorem for the big quantum cohomology of flag varieties, from the recent works on the small quantum cohomology mirror symmetry, via the equivariant unfolding theorem.

Unfolding of equivariant F-bundles and application to the mirror symmetry of flag varieties

TL;DR

This work introduces an equivariant unfolding framework for F-bundles, extending the Hertling-Manin universal unfolding to handle torus actions and infinite-rank settings. Central to the approach are the (IC), (GC), and (GC') conditions that guarantee existence and uniqueness of maximal unfoldings, together with framings that enable control of the u-direction. The authors apply this theory to flag varieties, constructing equivariant big and small quantum D-modules for G/P and proving big quantum cohomology mirror symmetry by unfolding the B-model Landau–Ginzburg data and relating it to the A-model via equivariant F-bundles. The results yield a precise isomorphism of big equivariant D-modules (and hence big quantum cohomology) for flag varieties, with a non-equivariant limit recovering the classical big-mirror statement. This provides a robust, reconstruction-based route to big-quantum mirror symmetry beyond the semisimple or -generated cases, and clarifies the role of equivariance in enabling universal unfoldings in Gromov–Witten theory.

Abstract

We establish an unfolding theorem for equivariant F-bundles (a variant of Frobenius manifolds), generalizing Hertling-Manin's universal unfolding of meromorphic connections. As an application, we obtain the mirror symmetry theorem for the big quantum cohomology of flag varieties, from the recent works on the small quantum cohomology mirror symmetry, via the equivariant unfolding theorem.
Paper Structure (24 sections, 30 theorems, 118 equations)

This paper contains 24 sections, 30 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main results
  4. Equivariant unfolding theorem
  5. Application to mirror symmetry of flag varieties
  6. (T)-structures and equivariant F-bundles
  7. Completions
  8. F-bundles and (T)-structures
  9. Lift of (T)-structures
  10. Equivariant F-bundles
  11. Unfolding of equivariant F-bundles
  12. Framing of (T)-structures
  13. Formal Hertling-Manin unfolding theorem
  14. Unfolding theorem for equivariant F-bundles
  15. Application to mirror symmetry of flag varieties
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\mathcal{F} = \lbrace (\mathcal{H} ,\nabla ) , (\mathcal{H}_R,\nabla_R) ,\alpha \rbrace$ be an equivariant F-bundle over $\Bbbk[\![\mathbf t_I]\!]$, and fix $v\in \mathcal{H}_R |_{u=t_I=0}$. Furthermore, any framing for $\mathcal{F}$ induces a unique framing on a maximal unfolding.

Theorems & Definitions (76)

  • Theorem 1.1: Unfolding of equivariant F-bundles, \ref{['thm:max-unfolding-arbitrary-rank']}
  • Proposition 1.2: \ref{['lemma:lift-T-structure-morphism-to-F-bundle']}
  • Theorem 1.3: Formal Hertling-Manin unfolding, \ref{['thm:Hertling-Manin-F-bundles']}
  • Theorem 1.4: Big quantum $D$-module mirror symmetry, Theorem \ref{['thm:big-mirror-symmetry-flag']}
  • Remark 2.2
  • Definition 2.3: F-bundle, (T)-structure
  • Definition 2.4: Framing
  • Lemma 2.6
  • proof
  • Remark 2.8
  • ...and 66 more