arXiv subjects

Tsukasa Ishibashi

Publications and source records attributed to Tsukasa Ishibashi.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic study of quantum configuration spaces of decorated flags

Let $G$ be a connected, simply connected complex simple algebraic group and $\mathscr{A}_G=G/U^+$ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ gives a quantum analogue of the configuration space $\mathrm{Conf}_K \mathscr{A}_G$ of $K$ decorated flags, which provides local building blocks for the Fock--Goncharov moduli space $\mathscr{A}_{G,Σ}$ of decorated twisted $G$-local systems on a marked surface $Σ$. We establish basic algebraic properties of $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ such as the domain property, quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of $\mathscr{A}_{G,Σ}$. Moreover, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ by transporting the Berenstein--Zelevinsky quantum cluster structure on $\mathcal{O}_q(G)$ via quantum Wilson lines, and prove that $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov--Shen exchange matrices. We also show that quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ restrict to those for $\mathcal{O}_q(\mathrm{Conf}_3 \mathscr{A}_G)$. Finally, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$, $K\geq 5$, and prove that the corresponding quantum cluster algebras contain the quantum configuration space $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$.

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arXiv2026

Adjoint Reidemeister torsion of 3-manifolds with torus boundary for semisimple algebraic groups

Let $M$ be a compact oriented $3$-manifold with boundary consisting of tori, and let $G$ be a semisimple algebraic group. We define the adjoint torsion function on the moduli stack of $G$-local systems on $M$ satisfying a certain regularity condition, extending the construction by Porti for $G = \mathrm{SL}_2$. When $M$ is a cusped hyperbolic manifold, we prove that the local system associated with the image of the complete hyperbolic structure via a principal embedding $\mathrm{PGL}_2 \to G$ satisfies the regularity condition. Moreover, we provide a formula expressing its adjoint torsion as a product of $\mathrm{PGL}_2$-torsions associated with the simple $\mathrm{PGL}_2$-modules with multiplicity given by the exponents of the Lie algebra of $G$. We compute the adjoint $\mathrm{PGSp}_4$-torsions of the figure-eight knot complement for two boundary-unipotent local systems, one is arising from the complete hyperbolic structure via a principal embedding, and the other is defined over a number field of degree $6$ and not arising from any $\mathrm{PGL}_2$-local system via principal embeddings.

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arXiv2026

Fixed point theorem for cluster modular groups

We prove that any finite subgroup $G \subset Γ_{\boldsymbol{s}}$ of the cluster modular group has fixed points in the cluster manifolds $\mathcal{A}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ and $\mathcal{X}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ under a certain condition. This generalizes Kerckhoff's Nielsen realization theorem [Ker83] for the mapping class group action on the Teichmüller space. The condition holds whenever $Γ_{\boldsymbol{s}}$ admits a cluster DT transformation, and it can be also verified for all finite mutation types except for $X_7$. Our proof closely follows Kerckhoff's argument, based on the convexity of log-cluster variables.

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arXiv2026

Cyclic quantum Teichmüller theory

Based on the pioneering ideas of Kashaev [Kas98,Kas00], we present a fully explicit construction of a finite-dimensional projective representation of the dotted Ptolemy groupoid when the quantum parameter $q$ is a root of unity, which reproduces the central charge of the $SU(2)$ Wess--Zumino--Witten model. A basic ingredient is the cyclic quantum dilogarithm [FK94]. A notable contribution of this work is a reinterpretation of the relations among the parameters in the cyclic quantum dilogarithm to ensure its pentagon identity in terms of the mutations of coefficients. In particular, we find the dual roles of these parameters: as coefficients in quantum cluster algebras and as the central characters of quantum cluster variables. We also provide a geometric method to decompose the space of quantum states into irreducible modules of the Chekhov--Fock algebra. We introduce two versions of quantum intertwiners associated with a mapping class: on the entire representation space and on each irreducible component, each being an explicit composite of cyclic quantum dilogarithm operators. We prove that the former gives an intertwiner of local representations of quantum Teichmüller space in the sense of Bai--Bonahon--Liu [BBL07], and also coincides with the transpose of the reduced quantum hyperbolic operator of Baseilhac--Benedetti [BB18]. The mutation relation of coefficients is equivalent to the quantum gluing equation. The irreducible intertwiner conjecturally coincides with the Bonahon--Liu intertwiner [BL07], and we give a partial evidence.

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arXiv2025

Bounded $\mathfrak{sp}_4$-laminations and their intersection coordinates

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 $\mathfrak{sp}_4$.

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arXiv2025

Quantum duality maps, skein algebras and their ensemble compatibility

We generalize the quantum duality map $\mathbb{I}_{\mathcal{A}}$ of Allegretti--Kim [AK17] 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 $\mathbb{I}_{\mathcal{X}}$ on the dual side based on the quantum bracelets basis [Thu14, MQ23]. Our construction factors through reduced stated skein algebras, based on the quantum trace maps [Lê18] together with an appropriate way of \emph{skein lifting} of integral $\mathcal{A}$-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.

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arXiv2024

A characterization of pseudo-Anosov mapping classes via tropical cluster transformations

We give a characterization of generic pseudo-Anosov mapping classes purely in terms of their expressions in the shear coordinates, thus giving an answer to a problem raised by Papadopoulos--Penner [PP93]. This characterization has a cluster algebraic generalization called the sign stability introduced in [IK21]. By combining with the results in [IK21], we see that the algebraic entropies of the cluster $\mathcal{A}$- and $\mathcal{X}$-transformations induced by a generic pseudo-Anosov mapping class both coincide with its topological entropy.

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arXiv2024

Earthquake theorem for cluster algebras of finite type

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.

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arXiv2024

Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$

Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$-case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $Σ$ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural $\mathbb{Z}_q$-form $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q} \subset \mathscr{S}_{\mathfrak{sp}_4,Σ}^q$, while the natural coefficient ring $\mathcal{R}$ of $\mathscr{S}_{\mathfrak{sp}_4,Σ}^q$ includes the inverse of the quantum integer $[2]_q$. We prove that its boundary-localization $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q}[\partial^{-1}]$ is included into a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sp}_4,Σ}$ that quantizes the function ring of the moduli space $\mathcal{A}_{Sp_4,Σ}^\times$. Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to [IY21](arXiv:2101.00643). We also propose a characterization of cluster variables in the spirit of Fomin--Pylyavksyy [FP16](arXiv:1210.1888) in terms of the $\mathfrak{sp}_4$-webs, and give infinitely many supporting examples on a quadrilateral.

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arXiv2024

Skein and cluster algebras with coefficients for unpunctured surfaces

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{Σ,\mathbb{W}}^{A}$ of a walled surface $(Σ,\mathbb{W})$, and prove that it has a quantum cluster structure. The walled surfaces naturally generalize the marked surfaces with multi-laminations, which have been used to describe the quantum cluster algebras of geometric type for marked surfaces by Fomin--Thurston [FT18]. Moreover, we give skein theoretic interpretation for some of quasi-homomorphisms [Fra16] between these quantum cluster algebras.

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arXiv2024

Unbounded $\mathfrak{sl}_3$-laminations around punctures

We continue to study the unbounded $\mathfrak{sl}_3$-laminations [IK22], with a focus on their structures at punctures. A key ingredient is their relation to the root data of $\mathfrak{sl}_3$. After giving a classification of signed $\mathfrak{sl}_3$-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 [SSW23] and Fraser--Pylyavskyy [FP21]. Finally, we discuss a formulation of unbounded $\mathfrak{g}$-laminations for a general semisimple Lie algebra $\mathfrak{g}$ in brief.

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arXiv2024

Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $\mathcal{X}_{\mathfrak{sl}_3,Σ}$, 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 $\mathcal{P}_{PGL_3,Σ}$ 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.

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arXiv2024

Entropy of cluster DT transformations and the finite-tame-wild trichotomy of acyclic quivers

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.

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arXiv2023

Wilson lines and their Laurent positivity

For a marked surface $Σ$ and a semisimple algebraic group $G$ of adjoint type, we study the Wilson line morphism $g_{[c]}:\mathcal{P}_{G,Σ} \to G$ associated with the homotopy class of an arc $c$ connecting boundary intervals of $Σ$, 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 $\mathcal{O}(\mathcal{P}_{G,Σ})$ when $Σ$ 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 $Σ$. 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 $Σ$, for suitable $f$ and $v$.

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arXiv2023

$\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems

For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $Σ$ with at least two marked points and no punctures, we prove that the cluster algebra $\mathscr{A}_{\mathfrak{g},Σ}$ associated with the pair $(\mathfrak{g},Σ)$ coincides with the upper cluster algebra $\mathscr{U}_{\mathfrak{g},Σ}$. The proof is based on the fact that the function ring $\mathcal{O}(\mathcal{A}^\times_{G,Σ})$ of the moduli space of decorated twisted $G$-local systems on $Σ$ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras $\mathscr{S}_{\mathfrak{g}, Σ}[\partial^{-1}]$ [Muller,IY23,IY22] for $\mathfrak{g}=\mathfrak{sl}_2, \mathfrak{sl}_3,$ or $\mathfrak{sp}_4$ are isomorphic to the cluster algebras $\mathscr{A}_{\mathfrak{g}, Σ}$.

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arXiv2023

Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sl}_3$

For an unpunctured marked surface $Σ$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},Σ}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $Σ$ with the boundary skein relations at marked points. We construct a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sl}_3,Σ}$ inside the skew-field $\mathrm{Frac}\mathscr{S}_{\mathfrak{sl}_{3},Σ}^{q}$ of fractions, which quantizes the cluster $K_2$-structure on the moduli space $\mathcal{A}_{SL_3,Σ}$ of decorated $SL_3$-local systems on $Σ$. We show that the cluster algebra $\mathscr{A}^q_{\mathfrak{sl}_3,Σ}$ contains the boundary-localized skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},Σ}^{q}[\partial^{-1}]$ as a subalgebra, and their natural structures, such as gradings and certain group actions, agree with each other. We also give an algorithm to compute the Laurent expressions of a given $\mathfrak{sl}_3$-web in certain clusters and discuss the positivity of coefficients. In particular, we show that the bracelets and the bangles along an oriented simple loop in $Σ$ have Laurent expressions with positive coefficients, hence give rise to quantum GS-universally positive Laurent polynomials.

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arXiv2023

Teichmüller and lamination spaces with pinnings

We describe the spaces of the positive and tropical points of the moduli space $\mathcal{P}_{PGL_2,Σ}$ introduced by Goncharov--Shen [GS19] as certain Teichmüller and lamination spaces, respectively, with additional data of pinnings. In the case where the surface $Σ$ has no punctures, we obtain the formulae relating various functions on the Teichmüller space with pinnings: $λ$-lengths, cross ratio coordinates, and Wilson lines. A topological description of the tropicalized amalgamation map is given in terms of $\mathcal{P}$-laminations. Based on our topological study of these "$\mathcal{P}$-type" spaces, we investigate the compatibility of the Fock--Goncharov duality maps $\mathbb{I}_\mathcal{A}$, $\mathbb{I}_\mathcal{X}$ constructed by [FG06,FG07,MSW13,GS15] under the extended ensemble map. We also discuss the amalgamation of bracelet bases.

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arXiv2021

Sign stability of mapping classes on marked surfaces II: general case via reductions

We give a cluster algebraic description of the reduction procedure of mapping classes along a multicurve. Based on this description, we characterize pseudo-Anosov mapping classes on a general marked surface in terms of a weaker version of the uniform sign stability, generalizing the main result in the previous paper [IK20]. Moreover we axiomatize a general reduction procedure of mutation loops parametrized by a rational polyhedral cone in the tropical cluster $\mathcal{X}$-variety, which includes both the reduction along a multicurve and the cluster reduction introduced in [Ish19].

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