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A mirror theorem for partial flag bundles

Ionut Ciocan-Fontanine, Yuki Koto

TL;DR

The paper proves a nonabelian mirror theorem for partial flag bundles, constructing a Weyl-invariant point on the genus-zero Gromov-Witten Lagrangian cone $\mathcal{L}_{Fl}$ from a Weyl-invariant $I$-function of an ambient prequotient. The strategy embeds a flag bundle into a trivial flag bundle as a zero locus, then reduces to twisted GW theory on an abelian quotient, applies an abelian/nonabelian correspondence, and uses quantum Riemann–Roch and differential-operator actions to complete the argument. The central result, Theorem mirror_Fl, generalizes earlier results for projective bundles and split flag bundles (IK and Oh) to the non-split setting and provides a framework for decomposing quantum cohomology via $\mathfrak{g}/\mathfrak{t}$-twisting. This work also lays groundwork for extending the abelian/nonabelian correspondence to arbitrary flag bundles and enables a deeper understanding of the Langrangian cone under nonabelian GIT quotients, with potential applications to decompositions of quantum $D$-modules and beyond.

Abstract

We construct a family of points on the Lagrangian cone of a partial flag bundle associated to a (possibly non-split) vector bundle from any Weyl-invariant $I$-function of a prequotient. This result can be seen as the nonabelian analogue of the mirror theorem for projective bundles in arXiv:2307.03696, and generalizes Oh's mirror theorem for split partial flag bundles in arXiv:1607.08326.

A mirror theorem for partial flag bundles

TL;DR

The paper proves a nonabelian mirror theorem for partial flag bundles, constructing a Weyl-invariant point on the genus-zero Gromov-Witten Lagrangian cone from a Weyl-invariant -function of an ambient prequotient. The strategy embeds a flag bundle into a trivial flag bundle as a zero locus, then reduces to twisted GW theory on an abelian quotient, applies an abelian/nonabelian correspondence, and uses quantum Riemann–Roch and differential-operator actions to complete the argument. The central result, Theorem mirror_Fl, generalizes earlier results for projective bundles and split flag bundles (IK and Oh) to the non-split setting and provides a framework for decomposing quantum cohomology via -twisting. This work also lays groundwork for extending the abelian/nonabelian correspondence to arbitrary flag bundles and enables a deeper understanding of the Langrangian cone under nonabelian GIT quotients, with potential applications to decompositions of quantum -modules and beyond.

Abstract

We construct a family of points on the Lagrangian cone of a partial flag bundle associated to a (possibly non-split) vector bundle from any Weyl-invariant -function of a prequotient. This result can be seen as the nonabelian analogue of the mirror theorem for projective bundles in arXiv:2307.03696, and generalizes Oh's mirror theorem for split partial flag bundles in arXiv:1607.08326.
Paper Structure (30 sections, 17 theorems, 86 equations)

This paper contains 30 sections, 17 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Genus zero Gromov--Witten theory
  3. Gromov--Witten invariant
  4. Gromov--Witten invariant
  5. torus-equivariant Gromov--Witten invariant
  6. twisted Gromov--Witten invariant
  7. Quantum cohomology
  8. Givental cone
  9. Givental cone for smooth projective varieties
  10. $\mathbb{T}$-equivariant Givental cone of smooth quasi-projective varieties
  11. twisted Givental cone
  12. $\mathbb{T}$-equivariant Givental cone of vector bundles
  13. Quantum Riemann-Roch theorem
  14. Differential operators acting on Givental cones
  15. Flag bundles, associated abelian quotients, and Weyl-invariant Givental cones
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $B$, $V$, and $r$ be as above. Let $\mathbb{T}=(\mathbb{C}^\times)^r$ act on $V^{\oplus r}$ by scaling each component diagonally, and let $\mathbb{W}:=\mathfrak{S}_r$ act on $V^{\oplus r}$ by permuting the components. Let $z\cdot I^{\lambda_\bullet}_{V^{\oplus r}}(x,z)$ be a $\mathbb{W}$-invaria where $q$ denotes the Novikov variable for the line in fibers, $H_1,\dots,H_r$ denote the Chern roo

Theorems & Definitions (32)

  • Theorem 1.1: Special case of Theorem \ref{['thm:mirror_Fl']}
  • Remark 1.2
  • Theorem 1.3: Non-equivariant version of Theorem \ref{['thm:abelian/nonabelian_correspondence']}
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 2.7: IK:quantum
  • ...and 22 more