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Crepant Transformation Conjecture For the Grassmannian Flop

Wendelin Lutz, Qaasim Shafi, Rachel Webb

TL;DR

The paper proves an explicit Crepant Transformation Conjecture for Grassmannian flops by relating two noncompact GIT quotients $X//_{ heta_+} G$ and $X//_{ heta_-} G$ through abelianization and reconstruction. It constructs a Weyl-anti-invariant symplectic isomorphism between Givental spaces and shows that big points on the associated Lagrangian cones are connected by analytic continuation, with two specialization paths for the Novikov parameter producing distinct continuations. The approach combines abelianization of quasimap $I$-functions, the method of big $I$-functions (explicit reconstruction), and a toric-wall-crossing framework to transfer toric results to a nonabelian Grassmannian setting; the techniques generalize to representations of groups isogenous to products of $SL(n)$ and relate to parallel toric and nonabelian wall-crossing literature. The Grassmannian flop thus provides a concrete, noncompact test case where equivariant, abelianization-based CTC can be established, highlighting the power of reconstruction and Weyl-anti-invariance in bridging nonabelian and toric data.

Abstract

We prove an explicit form of the Crepant Transformation Conjecture for Grassmannian flops. Our approach uses abelianization to first relate the restrictions of the Lagrangian cones to degree-2 classes, and then deduces the general result using ``explicit reconstruction'' (also known as the method of big I-functions).

Crepant Transformation Conjecture For the Grassmannian Flop

TL;DR

The paper proves an explicit Crepant Transformation Conjecture for Grassmannian flops by relating two noncompact GIT quotients and through abelianization and reconstruction. It constructs a Weyl-anti-invariant symplectic isomorphism between Givental spaces and shows that big points on the associated Lagrangian cones are connected by analytic continuation, with two specialization paths for the Novikov parameter producing distinct continuations. The approach combines abelianization of quasimap -functions, the method of big -functions (explicit reconstruction), and a toric-wall-crossing framework to transfer toric results to a nonabelian Grassmannian setting; the techniques generalize to representations of groups isogenous to products of and relate to parallel toric and nonabelian wall-crossing literature. The Grassmannian flop thus provides a concrete, noncompact test case where equivariant, abelianization-based CTC can be established, highlighting the power of reconstruction and Weyl-anti-invariance in bridging nonabelian and toric data.

Abstract

We prove an explicit form of the Crepant Transformation Conjecture for Grassmannian flops. Our approach uses abelianization to first relate the restrictions of the Lagrangian cones to degree-2 classes, and then deduces the general result using ``explicit reconstruction'' (also known as the method of big I-functions).
Paper Structure (30 sections, 24 theorems, 166 equations, 1 figure)

This paper contains 30 sections, 24 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Theorem statement
  4. Relationship to other work
  5. Summary of the paper
  6. Acknowledgements
  7. Background: equivariant Givental formalism for GIT quotients
  8. Poincaré pairings and Gromov-Witten invariants
  9. Givental space as a vector space
  10. Givental space as a formal scheme
  11. Equivariant Chow groups of GIT quotients
  12. Notation
  13. Nonequivariant limit
  14. Weyl action
  15. Equivariant Chow ring of $X\mathord{/\mkern-6mu/}_\theta G$
  16. ...and 15 more sections

Key Result

Theorem 1.2.1

There is a moduli space $\mathcal{M} \simeq \widetilde{\mathbb{P}^1}$, a degree-preserving symplectic isomorphism $\mathbb{U}: H_G^+ \to H_G^-$ of $F_K((z^{-1}))$-modules, and series such that

Figures (1)

  • Figure 1: The left diagram depicts the fan $\Sigma_T \subset \chi(T)_{\mathbb{R}}$ when $k=2$ (consisting of four 2-dimensional cones and their faces). The Weyl group $S_2$ acts by reflection over the diagonal, and $\Sigma_G$ is the fan pictured with two 1-dimensional cones and the origin. The right diagram is the toric variety $\mathcal{M}_T \simeq \mathbb{P}^1 \times \mathbb{P}^1$ with its four torus-invariant divisors and the Weyl-invariant subspace $\mathcal{M}_{G} \simeq \mathbb{P}^1$. The divisors labeled $\mathcal{M}_1$ and $\mathcal{M}_2$ will appear in the analytic continuation in Section \ref{['sec:cij']}.

Theorems & Definitions (61)

  • Theorem 1.2.1
  • Remark 1.2.2
  • Remark 1.2.3
  • Lemma 3.2.1
  • proof
  • Lemma 3.3.1
  • proof
  • Proposition 3.4.1
  • Lemma 3.4.2
  • proof
  • ...and 51 more