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Scattering diagrams and Jeffrey-Kirwan residues

Sara Angela Filippini, Jacopo Stoppa

Abstract

We show that the consistent completion of an initial scattering diagram in $M_{\mathbb{R}}$ (for a finite rank lattice $M$) can be expressed quite generally in terms of the Jeffrey-Kirwan residues of certain explicit meromorphic forms, by using the Maurer-Cartan asymptotic analysis developed by Chan-Leung-Ma and Leung-Ma-Young. A similar result holds for the associated theta functions.

Scattering diagrams and Jeffrey-Kirwan residues

Abstract

We show that the consistent completion of an initial scattering diagram in (for a finite rank lattice ) can be expressed quite generally in terms of the Jeffrey-Kirwan residues of certain explicit meromorphic forms, by using the Maurer-Cartan asymptotic analysis developed by Chan-Leung-Ma and Leung-Ma-Young. A similar result holds for the associated theta functions.
Paper Structure (20 sections, 13 theorems, 131 equations)

This paper contains 20 sections, 13 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Relation to enumerative geometry
  4. Theta functions
  5. Possible developments
  6. Maurer-Cartan scattering
  7. Unfolding
  8. MC scattering and (higher) residue pairing
  9. Behaviour of the oscillatory integral
  10. The case $p \in \operatorname{Int}_{re}(\Gamma) = \Gamma\setminus\partial\Gamma$
  11. The case $p \in \partial\Gamma$
  12. The case $p \in \operatorname{Lie}(\mathbb{T}_{ \mathcal{L}}) \setminus \Gamma$
  13. Unfolding
  14. JK residues
  15. Global residue
  16. ...and 5 more sections

Key Result

Theorem 1.3

The coefficient $c_m$ of the consistent diagram $\mathfrak{D}_{c}$ along the wall $P$, in the fixed degree $m \in M^+_{\sigma}$, is given by the transversal integral of the Jeffrey-Kirwan residue $1$-form of $Z^{(P, m)}_{c, \delta}(s)$, namely where $\operatorname{JK}$ is taken with respect to the complex variables $s = (s_1, \ldots, s_K)$ and the set of affine hyperplanes while the chamber $\xi

Theorems & Definitions (29)

  • Example 1.1: KimUeda, Section 2
  • Example 1.2: beau, Section 3
  • Theorem 1.3: Theorem \ref{['MainThm']}
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Proposition \ref{['ThetaProp']}
  • Remark 1.7
  • Example 2.1
  • Remark 2.2
  • Theorem 2.3: leung, Theorem 3.12 and Proposition 3.14
  • ...and 19 more