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Real algebraic curves and twisted Hurwitz numbers

Yurii Burman, Raphaël Fesler

Abstract

We provide a direct correspondence between the $b$-Hurwitz numbers with $b=1$ from \cite{ChapuyDolega}, and twisted Hurwtiz numbers from \cite{TwistedHurwitz}. This provides a description of real coverings of the sphere with ramification on the real line in terms of monodromy.

Real algebraic curves and twisted Hurwitz numbers

Abstract

We provide a direct correspondence between the -Hurwitz numbers with from \cite{ChapuyDolega}, and twisted Hurwtiz numbers from \cite{TwistedHurwitz}. This provides a description of real coverings of the sphere with ramification on the real line in terms of monodromy.
Paper Structure (6 sections, 19 theorems, 3 equations, 6 figures)

This paper contains 6 sections, 19 theorems, 3 equations, 6 figures.

Table of Contents

  1. Preliminaries
  2. A combinatorial correspondence
  3. An analytico-geometric correspondence
  4. Decorated Hurwitz space
  5. Lyashko--Looijenga map and real Hurwitz space
  6. From geometry to combinatorics and vice versa

Key Result

Proposition 1.1

$B_\lambda^{\sim}$ is a $B_n$-conjugacy class in $S_{2n}$.

Figures (6)

  • Figure 1: Label Correspondence
  • Figure 2: $\Lambda(\delta_{i}, \delta_{i+1})=[2,1^{n-2}]$
  • Figure 3: Perfect Matching
  • Figure 4: Monodromy
  • Figure 5: The loop $\delta_{k-1}\delta_k$
  • ...and 1 more figures

Theorems & Definitions (32)

  • Proposition 1.1: TwistedHurwitz
  • Definition 1.2
  • Definition 1.3: ChapuyDolega
  • Theorem 1.4: ChapuyDolega
  • Definition 1.5
  • Proposition 1.6
  • Corollary 1.7: of Theorem \ref{['th:CDConstel']} and Proposition \ref{['Pp:LabelToCDLabel']}
  • Theorem 1.8: BenDali
  • Corollary 1.9: BenDali
  • Proposition 2.1
  • ...and 22 more