Tsukasa Ishibashi

We introduce rational bounded $\mathfrak{sp}_4$-laminations on a marked surface $\boldsymbolΣ$ as a proposed topological model for the rational tropical points $\mathcal{A}_{Sp_4,\boldsymbolΣ}(\mathbb{Q}^{\mathsf{T}})$ of the Fock--Goncharov moduli space [FG06]. Our space consists of certain equivalence classes of $\mathfrak{sp}_4$-webs introduced by Kuperberg [Kup96], together with rational measures. We define tropical coordinate systems using the $\mathfrak{sp}_4$-case of the intersection number of Shen--Sun--Weng [SSW25], and establish a bijection using the framework of the graded $\mathfrak{sp}_4$-skein algebra. This provides a topological perspective for Fock--Goncharov duality for $\mathfrak{sp}_4$.

We continue to study the unbounded $$\mathfrak {sl}_3$$ sl 3 -laminations [16], with a focus on their structures at punctures. A key ingredient is their relation to the root data of $$\mathfrak {sl}_3$$ sl 3 . After giving a classification of signed $$\mathfrak {sl}_3$$ sl 3 -webs around a puncture, we describe the tropicalization of the Goncharov–Shen’s Weyl group action in detail. We also clarify the relationship with several other approaches by Shen–Sun–Weng [28] and Fraser–Pylyavskyy [10]. Finally, we discuss a formulation of unbounded $$\mathfrak {g}$$ g -laminations for a general semisimple Lie algebra $$\mathfrak {g}$$ g in brief.

We generalize the quantum duality map $$\mathbb {I}_{\mathcal {A}}$$I A of Allegretti–Kim (Adv Math 306:1164–1208, 2017) for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map $$\mathbb {I}_{\mathcal {X}}$$I X on the dual side based on the quantum bracelets basis (Mandel and Qin in arXiv:2301.11101; Thurston in Proc Natl Acad Sci USA 111(27):9725–9732, 2014). Our construction factors through reduced stated skein algebras, based on the quantum trace maps (Lê in Quantum Topol 9(3):591–632, 2018) together with an appropriate way of skein lifting of integral $$\mathcal {A}$$A -laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.

Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $\mathcal{X}_{\mathfrak{sl}_3,\Sigma}$, which we call unbounded $\mathfrak{sl}_3$-laminations, based on the Kuperberg's $\mathfrak{sl}_3$-webs. We introduce their tropical cluster coordinates as an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any ideal triangulation. As a tropical analogue of gluing morphisms among the moduli spaces $\mathcal{P}_{PGL_3,\Sigma}$ of Goncharov--Shen, we describe a geometric gluing procedure of unbounded $\mathfrak{sl}_3$-laminations with pinnings via ``shearings''. We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra [IY23], which conjecturally leads to a quantum duality map.

We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the \emph{cluster earthquake map}. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of $\mathbb{R}^\mathrm{trop}$- and $\mathbb{R}_{>0}$-valued points of the cluster $\mathcal{X}$-variety. For those of type $A_n$ and $D_n$, the cluster earthquake map indeed recovers the earthquake maps for marked disks and once-punctured marked disks, respectively. Moreover, we investigate certain asymptotic behaviors of the cluster earthquake map, which give rise to "continuous deformations" of the Fock--Goncharov fan.

The cluster algebra associated with an acyclic quiver has a special mutation loop $τ$, called the cluster Donaldson--Thomas (DT) transformation, related to the Auslander--Reiten translation. In this paper, we characterize the finite-tame-wild trichotomy for acyclic quivers by the sign stability of $τ$ introduced in [IK21] and its cluster stretch factor. As an application, we compute several kinds of entropies of $τ$ and other mutation loops. In particular, we show that the algebraic and categorical entropies of $τ$ are commonly given by the logarithm of the spectral radius of the Coxeter matrix associated with the quiver, and that any mutation loop of finite or tame acyclic quivers have zero algebraic entropy.

For a marked surface $\Sigma$ and a semisimple algebraic group $G$ of adjoint type, we study the Wilson line morphism $g_{[c]}:\mathcal{P}_{G,\Sigma} \to G$ associated with the homotopy class of an arc $c$ connecting boundary intervals of $\Sigma$, which is the comparison element of pinnings via parallel-transport. The matrix coefficients of the Wilson lines give a generating set of the function algebra $\mathcal{O}(\mathcal{P}_{G,\Sigma})$ when $\Sigma$ has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov--Shen [GS19], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of $\Sigma$. We show that the matrix coefficients $c_{f,v}^V(g_{[c]})$ give Laurent polynomials with positive integral coefficients in the Goncharov--Shen coordinate system associated with any decorated triangulation of $\Sigma$, for suitable $f$ and $v$.

For a marked surface $$\Sigma $$Σ and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism $$g_{[c]}:{\mathcal {P} }_{G,\Sigma } \rightarrow G$$g [ c ] : P G , Σ → G associated with the homotopy class of an arc c connecting boundary intervals of $$\Sigma $$Σ , which is the comparison element of pinnings via parallel-transport. The matrix coefficients of the Wilson lines give a generating set of the function algebra $$\mathcal {O}({\mathcal {P} }_{G,\Sigma })$$O ( P G , Σ ) when $$\Sigma $$Σ has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov–Shen [18], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of $$\Sigma $$Σ . We show that the matrix coefficients $$c_{f,v}^V(g_{[c]})$$c f , v V ( g [ c ] ) give Laurent polynomials with positive integral coefficients in the Goncharov–Shen coordinate system associated with any decorated triangulation of $$\Sigma $$Σ , for suitable f and v.