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Wall Crossing and the Fourier-Mukai Transform for Grassmann Flops

Nathan Priddis, Mark Shoemaker, Yaoxiong Wen

TL;DR

This paper advances the crepant transformation program by proving a wall-crossing formula for relative Grassmann flops over a smooth base, connecting the generating functions of genus-zero Gromov–Witten invariants through analytic continuation and a linear symplectic transform. The approach hinges on the abelian/non-abelian correspondence to reduce to abelian quotients, computes explicit $I$-functions for Grassmann bundles, and constructs a symplectic operator $ ext{U}$ that intertwines $I$-functions of the two sides while preserving Iritani's integral structure via Fourier--Mukai in $K$-theory. A key technical achievement is the homotopy of the analytic-continuation path and a detailed combinatorial identity ensuring compatibility with the Fourier--Mukai kernel. The results extend the crepant-transformation framework to non-abelian variation of GIT quotients and fibered settings over arbitrary bases, aligning with and generalizing prior toric/abelian cases and parallel work in Lutz–Shafi–Webb. Overall, the work provides a conceptual and computational blueprint for higher-rank wall-crossings and their $K$-theoretic incarnations in Gromov–Witten theory.

Abstract

We prove the crepant transformation conjecture for relative Grassmann flops over a smooth base $B$. We show that the $I$-functions of the respective GIT quotients are related by analytic continuation and a symplectic transformation. We verify that the symplectic transformation is compatible with Iritani's integral structure, that is, that it is induced by a Fourier-Mukai transform in $K$-theory.

Wall Crossing and the Fourier-Mukai Transform for Grassmann Flops

TL;DR

This paper advances the crepant transformation program by proving a wall-crossing formula for relative Grassmann flops over a smooth base, connecting the generating functions of genus-zero Gromov–Witten invariants through analytic continuation and a linear symplectic transform. The approach hinges on the abelian/non-abelian correspondence to reduce to abelian quotients, computes explicit -functions for Grassmann bundles, and constructs a symplectic operator that intertwines -functions of the two sides while preserving Iritani's integral structure via Fourier--Mukai in -theory. A key technical achievement is the homotopy of the analytic-continuation path and a detailed combinatorial identity ensuring compatibility with the Fourier--Mukai kernel. The results extend the crepant-transformation framework to non-abelian variation of GIT quotients and fibered settings over arbitrary bases, aligning with and generalizing prior toric/abelian cases and parallel work in Lutz–Shafi–Webb. Overall, the work provides a conceptual and computational blueprint for higher-rank wall-crossings and their -theoretic incarnations in Gromov–Witten theory.

Abstract

We prove the crepant transformation conjecture for relative Grassmann flops over a smooth base . We show that the -functions of the respective GIT quotients are related by analytic continuation and a symplectic transformation. We verify that the symplectic transformation is compatible with Iritani's integral structure, that is, that it is induced by a Fourier-Mukai transform in -theory.
Paper Structure (24 sections, 22 theorems, 130 equations, 1 figure)

This paper contains 24 sections, 22 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Grassmann flops
  3. Statement of theorem
  4. Strategy of proof
  5. Relation to other work
  6. Geometric setup
  7. Cohomology
  8. The associated abelian quotients
  9. Fourier--Mukai transform
  10. Gromov--Witten theory and $\boldsymbol{I}$-functions
  11. Integral lattice
  12. $\boldsymbol{I}$-functions for Grassmann bundles
  13. $\boldsymbol{r=1}$
  14. The associated abelian quotients
  15. The non-abelian quotients
  16. ...and 9 more sections

Key Result

Theorem 1.2

There exists a linear symplectic isomorphism $\mathbb{U}\colon \widetilde{\mathcal{H}}_{X_{-}} \to \widetilde{\mathcal{H}}_{X_{+}}$ such that the following diagram commutes: \begin{tikzcd} K^0_\torus(X_{-}) \ar[r, "\FM"] \ar[d, "\Psi_-"] & K^0_\torus(X_{+}) \ar[d, "\Psi_+"] \\ \widetilde{\cc H}_{X_{ where $\widetilde{I_{X_{+}}}$ denotes the analytic continuation of $I_{X_{+}}$ along $\hat{\gamma}.

Figures (1)

  • Figure 3: The (real part of the) original path of analytic continuation $\gamma_1 \star \gamma_2$ (in purple) and the new path $\hat{\gamma}$ (in blue).

Theorems & Definitions (32)

  • Example 1.1
  • Theorem 1.2: Theorem \ref{['t:st3']}
  • Lemma 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Remark 2.8
  • ...and 22 more