اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدModuli spaces of real projective structures on surfaces: Notes on a paper by V.V. Fock and A.B. Goncharov
71
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharovs description of the moduli space of framed marked properly convex projective structures with minimal or maximal ends, and deduce results of Marquis and Goldman as consequences. We also discuss the Poisson structure on moduli space and its relationship to Goldmans Poisson structure on the character variety.
قيم البحث
اقرأ أيضاً
Harmonic maps that minimise the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences for the beha
viour of lumps and their symmetries. An interesting feature is that the moduli space of charge three lumps is a $7$-dimensional manifold of cohomogeneity one which can be described as a one-parameter family of symmetry orbits of $D_2$ symmetric maps. In this paper, we discuss the charge three moduli spaces of lumps from two perspectives: discrete symmetries of lumps and the Riemann-Hurwitz formula. We then calculate the metric and find explicit formulas for various geometric quantities. We also discuss the implications for lump decay.
For a finite-type surface $mathfrak{S}$, we study a preferred basis for the commutative algebra $mathbb{C}[mathcal{X}_{mathrm{SL}_3(mathbb{C})}(mathfrak{S})]$ of regular functions on the $mathrm{SL}_3(mathbb{C})$-character variety, introduced by Siko
ra-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $mathfrak{S}$. We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety.
In a companion paper (arXiv 2011.01768) we constructed non-negative integer coordinates $Phi_mathcal{T}$ for a distinguished collection $mathcal{W}_{3, widehat{S}}$ of $mathrm{SL}_3$-webs on a finite-type punctured surface $widehat{S}$, depending on
an ideal triangulation $mathcal{T}$ of $widehat{S}$. We prove that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $mathcal{T}^prime$ is chosen then the coordinate change map relating $Phi_mathcal{T}$ and $Phi_{mathcal{T}^prime}$ is a prescribed tropical cluster transformation. Moreover, when $widehat{S}=Box$ is an ideal square, we provide a topological geometric description of the Hilbert basis (in the sense of linear programming) of the non-negative integer cone $Phi_mathcal{T}(mathcal{W}_{3, Box}) subset mathbb{Z}_{geq 0}^{12}$, and we prove that this cone canonically decomposes into 42 sectors corresponding topologically to 42 families of $mathrm{SL}_3$-webs in the square.
This is an expository proof that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on $M$ is a subset of $operatorname{Hom}(pi_1M,operatorname{PGL}(n+1,mathbb RR))$ that is both open and closed.
Fock and Goncharov introduced cluster ensembles, providing a framework for coordinates on varieties of surface representations into Lie groups, as well as a complete construction for groups of type $A_n$. Later, Zickert, Le, and Ip described, using d
iffering methods, how to apply this framework for other Lie group types. Zickert also showed that this framework applies to triangulated $3$-manifolds. We present a complete, general construction, based on work of Fomin and Zelevinsky. In particular, we complete the picture for the remaining cases: Lie groups of types $F_4$, $E_6$, $E_7$, and $E_8$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
