Dynamics of Fricke–Painlevé VI Surfaces
URL: https://www.mdpi.com/2673-8716/4/1/1
Title: Dynamics of Fricke–Painlevé VI Surfaces
Author(s): Planat, M., Chester, D., & Irwin, K.
DOI: https://doi.org/10.3390/dynamics4010001
Publication Date: 2 January 2024
Resource Type: Link
Format: Research paper
Working Group: WG1-WG3
Affiliation(s): 1) CNRS, Institut FEMTO-ST, Université de Franche-Comté, F-25044 Besançon, France; 2) Quantum Gravity Research, Los Angeles, CA 90290, USA
Access Status: Open
Keywords:
Description: The symmetries of a Riemann surface Σ∖{𝑎𝑖} with n punctures 𝑎𝑖 are encoded in its fundamental group 𝜋1(Σ). Further structure may be described through representations (homomorphisms) of 𝜋1 over a Lie group G as globalized by the character variety 𝒞=Hom(𝜋1,𝐺)/𝐺. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere Σ=𝑆(4)2 and the ‘space-time-spin’ group 𝐺=𝑆𝐿2(ℂ). In such a situation, 𝒞 possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface 𝑉𝑎,𝑏,𝑐,𝑑(𝑥,𝑦,𝑧) with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or 𝑃𝑉𝐼); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of 𝑃𝑉𝐼. In this paper, we feature the parametric representation of some solutions of 𝑃𝑉𝐼: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups 𝑓𝑝 encountered in TQC or DNA/RNA sequences are proposed.
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