论文标题
$ {\ rm sl} _3 $ - 作为$ {\ rm pgl} _3 $ cluster for Surfaces的基础
${\rm SL}_3$-laminations as bases for ${\rm PGL}_3$ cluster varieties for surfaces
论文作者
论文摘要
在本文中，我们部分解决了Fock-Goncharov的二重性猜想，用于与其模量$ {\ rm g} $相关的聚类品种，当时是$ {\ rm g} $ a_2 $ a_2 $ a _2 $ a _2 $ { pgl} _3 $。基于Kuperberg的$ {\ rm sl} _3 $ -Webs，我们介绍了$ {\ rm sl} _3 $ - laminations $ \ frak {s} $定义为某些$ {\ rm sl} _3 $ webs的概念。我们介绍$ {\ rm sl} _3 $ - 层次的坐标系统，并证明满足一致性属性的$ {\ rm sl} _3 $ - 层次是群集的热带整数点的几何实现。 sl} _3，\ frak {s}} $。每个这样的$ {\ rm sl} _3 $ - 层次，我们在群集$ \ mathscr {x} $ - moduli space $ \ mathscr {x} _ {{\ rm pgl} _3，_3，\ frak {s}} $上构建常规功能。我们表明，这些功能构成了所有常规函数的环的基础。为了证明证明，我们为任何三角形边框表面和带有标记的点和状态和状态的公式开发了$ {\ rm sl} _3 $量子和经典的跟踪图。我们在$ \ mathscr {x} _ {{\ rm pgl} _3，\ frak {s}} $上构造基本常规函数的量子版本。本文构建的基础是由非纤维化网构建的，因此可以看作是更高的“手镯”基础，相应的“手镯”版本也可以被视为Fock-Goncharov和Allegretti-kim基础的直接类似物，用于$ {\ rm sl} _2 $ _2 $ - $ - $ - $ - $ - $ {
In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of ${\rm G}$-local systems on a punctured surface $\frak{S}$ with boundary data, when ${\rm G}$ is a group of type $A_2$, namely ${\rm SL}_3$ and ${\rm PGL}_3$. Based on Kuperberg's ${\rm SL}_3$-webs, we introduce the notion of ${\rm SL}_3$-laminations on $\frak{S}$ defined as certain ${\rm SL}_3$-webs with integer weights. We introduce coordinate systems for ${\rm SL}_3$-laminations, and show that ${\rm SL}_3$-laminations satisfying a congruence property are geometric realizations of the tropical integer points of the cluster $\mathscr{A}$-moduli space $\mathscr{A}_{{\rm SL}_3,\frak{S}}$. Per each such ${\rm SL}_3$-lamination, we construct a regular function on the cluster $\mathscr{X}$-moduli space $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$. We show that these functions form a basis of the ring of all regular functions. For a proof, we develop ${\rm SL}_3$ quantum and classical trace maps for any triangulated bordered surface with marked points, and state-sum formulas for them. We construct quantum versions of the basic regular functions on $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$. The bases constructed in this paper are built from non-elliptic webs, hence could be viewed as higher `bangles' bases, and the corresponding `bracelets' versions can also be considered as direct analogs of Fock-Goncharov's and Allegretti-Kim's bases for the ${\rm SL}_2$-${\rm PGL}_2$ case.
