[Paper Review] Finite braid group orbits in Aff(C)-character varieties of the punctured sphere

Gaël Cousin, Delphine Moussard|arXiv (Cornell University)|Apr 14, 2016

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper provides a complete classification of finite braid group orbits in $\mathrm{Aff}(\mathbb{C})$-character varieties of the punctured Riemann sphere for $n \geq 4$ punctures, using coalescence and finite complex reflection groups. The key result identifies precisely which linear monodromy parts yield finite orbits, leading to new characterizations of algebraic solutions to Garnier systems and connections to $F_D$-type Lauricella hypergeometric functions.

ABSTRACT

We give a complete description of finite braid group orbits in Aff(C)-character varieties of the punctured Riemann sphere. This is performed thanks to a coalescence procedure and to the theory of finite complex reflection groups. We then derive consequences in the theory of differential equations. These concern algebraicity of isomonodromic deformations for reducible rank two logarithmic connections on the sphere, the Riemann-Hilbert problem and F_D-type Lauricella hypergeometric functions.

Motivation & Objective

• To classify all finite orbits of the pure braid group action on $\mathrm{Aff}(\mathbb{C})$-character varieties of the $n$-punctured Riemann sphere for $n \geq 4$.
• To establish a connection between finite braid group orbits and algebraizable isomonodromic deformations of rank 2 reducible logarithmic connections on $\mathbb{CP}^1$.
• To derive new characterizations of algebraic solutions to Garnier systems and $F_D$-type Lauricella hypergeometric functions via monodromy group finiteness.
• To provide a systematic study of finite complex reflection groups $G_{25}$ and $G_{32}$ in the context of character varieties and their dynamics on $\mathbb{C}\mathbb{P}^2$ and $\mathbb{C}\mathbb{P}^3$.
• To demonstrate a conjugation between the flat connection from Schlesinger deformation and the one describing $F_D$-hypergeometric functions, recovering and refining Schwarz's list for $F_D$-systems.

Proposed method

• Use of a coalescence procedure to reduce the study of $n$-punctured cases to lower $n$, starting from $n=4$.
• Reduction of the braid group action to linear dynamics parametrized by the linear part of affine representations, fixing the linear component.
• Application of Shephard-Todd classification of finite complex reflection groups to analyze orbit finiteness in $\mathbb{C}\mathbb{P}^2$ and $\mathbb{C}\mathbb{P}^3$ for $G_{25}$ and $G_{32}$.
• Construction of a constant matrix $G = K + L + M$ to conjugate the flat connection $d_\theta$ to the $F_D$-hypergeometric system, with explicit formulas for $K$, $L$, and $M$.
• Use of the determinant formula $\det(G) = (-1)^{N+1}\theta_1(\alpha\theta_{N+1})^N$ to verify the conjugation and analyze monodromy finiteness.
• Projective monodromy correspondence between the Schlesinger system and $F_D$-hypergeometric systems, enabling recovery of Schwarz-type conditions.

Experimental results

Research questions

• RQ1 Which linear monodromy parts in $\mathrm{Aff}(\mathbb{C})$-representations of the fundamental group of the $n$-punctured sphere yield finite braid group orbits?
• RQ2 How can the coalescence procedure be used to reduce the classification of finite orbits from $n > 4$ to lower $n$, particularly $n=4$?
• RQ3 What is the precise correspondence between finite branching of $F_D$-hypergeometric functions and the finiteness of braid group orbits in the character variety?
• RQ4 How does the conjugation between the Schlesinger connection and the $F_D$-hypergeometric system allow for a new derivation of Schwarz’s list for $F_D$-systems?
• RQ5 What are the conditions on parameters $\alpha, \beta_1, \ldots, \beta_N, \gamma$ under which $F_D$-hypergeometric functions have finite branching, and how do they relate to monodromy finiteness?

Key findings

• For $n=4$, finite braid group orbits exist precisely when the linear part generates a finite subgroup of $\mathrm{GL}_2(\mathbb{C})$, as classified by Schwarz.
• The coalescence method allows the complete classification of finite orbits for $n \geq 4$ by reducing to the $n=4$ case and analyzing only finitely many linear groups.
• Finite orbits in $\mathbb{C}\mathbb{P}^2$ and $\mathbb{C}\mathbb{P}^3$ are fully classified for the complex reflection groups $G_{25}$ and $G_{32}$, respectively.
• The paper provides an explicit conjugation matrix $G = K + L + M$ that relates the flat connection from Schlesinger deformation to the $F_D$-hypergeometric system, with $\det(G) = (-1)^{N+1}\theta_1(\alpha\theta_{N+1})^N$.
• Theorem 6.3.1 establishes that for $N > 1$, $F_D$-hypergeometric functions have finite branching if and only if: (a) $N=2$ and four of the $\theta_i$ are $\pm\frac{1}{6}$ mod $\mathbb{Z}$, or (b) $N=3$ and all six $\theta_i$ are $\pm\frac{1}{6}$ mod $\mathbb{Z}$.
• The study recovers and refines Schwarz’s list for irreducible $F_D$-hypergeometric functions, with additional data on translation parts and orbit structure for reducible cases.