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On wall-crossing coordinates in Cerf theory

Roger Casals

TL;DR

The paper builds a precise bridge between wall-crossing coordinates in Cerf theory and cluster coordinates on braid varieties by expressing cluster variables as signed products of Morse-theoretic Bruhat numbers. Using an explicit pinning of SL$_n$, matrix factorizations, and principal-minor identities, it proves that each cluster variable $A_k$ equals a signed product of Bruhat numbers $eta_j(f_{x_k})$ evaluated after the $k$-th crossing. The argument translates Morse-differential changes under handleslides and crossings into determinant/minor relations among the P- and B-matrices, connecting to generalized minors in the braid-variety cluster algebra. The paper supplements the theory with concrete 2-, 3-, and 4-stranded examples and discusses broader connections to Legendrian generating families and possible categorical formalisms, indicating a rich interaction between Cerf theory, wall-crossing, and cluster algebras.

Abstract

We relate Bruhat numbers in real Morse theory to cluster variables in braid varieties. This provides instances of wall-crossing coordinates in the study of Cerf diagrams.

On wall-crossing coordinates in Cerf theory

TL;DR

The paper builds a precise bridge between wall-crossing coordinates in Cerf theory and cluster coordinates on braid varieties by expressing cluster variables as signed products of Morse-theoretic Bruhat numbers. Using an explicit pinning of SL, matrix factorizations, and principal-minor identities, it proves that each cluster variable equals a signed product of Bruhat numbers evaluated after the -th crossing. The argument translates Morse-differential changes under handleslides and crossings into determinant/minor relations among the P- and B-matrices, connecting to generalized minors in the braid-variety cluster algebra. The paper supplements the theory with concrete 2-, 3-, and 4-stranded examples and discusses broader connections to Legendrian generating families and possible categorical formalisms, indicating a rich interaction between Cerf theory, wall-crossing, and cluster algebras.

Abstract

We relate Bruhat numbers in real Morse theory to cluster variables in braid varieties. This provides instances of wall-crossing coordinates in the study of Cerf diagrams.
Paper Structure (13 sections, 1 theorem, 29 equations, 2 figures)

This paper contains 13 sections, 1 theorem, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scientific context
  3. Main result
  4. The argument
  5. Initial setup
  6. An explicit pinning
  7. Coordinatizing Morse Differentials
  8. Proof of the \ref{['thm:main_result_equality_DO_NOT_CREF']}
  9. A final comment
  10. A few examples
  11. 2-stranded case
  12. A 3-stranded example
  13. A 4-stranded example

Key Result

Lemma 2.4

Let $\beta=\sigma_{i_1}\sigma_{i_2}\cdots\sigma_{i_l}$ and $(\pi_\beta,\{f_x\})$ its associated Cerf diagram in normal form. Consider the Morse differential $\partial_k$ of a Morse function $f_k$ right after the $k$th crossing, and its Bruhat numbers $\bbbeta_j(f_k)$, $j\in[1,n]$. Then if $P_{i_k}(z_{i_k})\cdots P_{i_1}(z_1)w_0\in \mathsf{U} w_0\mathrm{T}\mathsf{U}$ lies in the unipotent Bruhat c

Figures (2)

  • Figure 1: (Left) The general form of a Cerf diagram $\pi_\beta$ associated to a positive braid word $\beta$. Each positive crossing of the braid word corresponds to a value exchange between critical points. (Right) An example with $\beta=(\sigma_1\sigma_2\sigma_3)^2 \sigma_1\sigma_2\sigma_1\sigma_3(\sigma_1\sigma_2\sigma_1)^2$.
  • Figure 2: A depiction of \ref{['eq:differentials_handleslides']} on the left, \ref{['eq:differentials_crossing']} at the center, and \ref{['eq:differentials_handleslide_crossing']} on the right. The Cerf diagram is in black, the handleslide marks in dashed blue, and the location and relation between the Morse differentials in gray.

Theorems & Definitions (8)

  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Lemma 2.4: Bruhat numbers from handleslide marks
  • proof
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1