arXiv subjects

Hyun Kyu Kim

Publications and source records attributed to Hyun Kyu Kim.

At least 19 recordsLinked to original sources

arXiv2026

Naturality of ${\rm SL}_n$ quantum trace maps for surfaces

The ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$, studied by Sikora, is an algebra generated by isotopy classes of $n$-webs living in the thickened surface $\mathfrak{S} \times (-1,1)$, where an $n$-web is a union of framed links and framed oriented $n$-valent graphs satisfying certain conditions. For each ideal triangulation $λ$ of $\mathfrak{S}$, Lê and Yu constructed an algebra homomorphism, called the ${\rm SL}_n$-quantum trace, from the ${\rm SL}_n$-skein algebra of $\mathfrak{S}$ to a so-called balanced subalgebra of the $n$-root version of Fock and Goncharov's quantum torus algebra associated to $λ$. We show that the ${\rm SL}_n$-quantum trace maps for different ideal triangulations are related to each other via a balanced $n$-th root version of the quantum coordinate change isomorphism, which extends Fock and Goncharov's isomorphism for quantum cluster varieties. We avoid heavy computations in the proof, by using the splitting homomorphisms of Lê and Sikora, and a network dual to the $n$-triangulation of $λ$ studied by Schrader and Shapiro.

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arXiv2026

Quantized Geodesic Lengths for Teichmüller Spaces: Algebraic Aspects

In 1980's H. Verlinde suggested to construct and use a quantization of Teichmüller spaces to construct spaces of conformal blocks for the Liouville conformal field theory. This suggestion led to a mathematical formulation by Fock in 1990's and later by Fock, Goncharov and Shen, called the modular functor conjecture, based on the Chekhov-Fock quantum Teichmüller theory. In 2000's, Teschner combined the Chekhov-Fock version and the Kashaev version of quantum Teichmüller theory to construct a solution to a modified form of the conjecture. We embark on a direct approach to the conjecture based on the Chekhov-Fock(-Goncharov) theory. We construct quantized trace-of-monodromy along simple loops via Bonahon and Wong's quantum trace maps developed in 2010's, and investigate algebraic structures of them, which will eventually lead to construction and properties of quantized geodesic length operators. We show that a special recursion relation used by Teschner is satisfied by the quantized trace-of-monodromy, and that the quantized trace-of-monodromy for disjoint loops commute in a certain strong sense.

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arXiv2025

Monoidal categorification of genus zero skein algebras

We prove a conjecture of the first and third named authors relating the Kauffman bracket skein algebra of a genus zero surface with boundary to a quantized $K$-theoretic Coulomb branch. As a consequence, we see that our skein algebra arises as the Grothendieck ring of the bounded derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima variety of triples with monoidal structure defined by the convolution product. We thus give a monoidal categorification of the skein algebra, partially answering a question posed by D. Thurston.

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arXiv2025

Frobenius homomorphisms for stated ${\rm SL}_n$-skein modules

The stated ${\rm SL}_n$-skein algebra $\mathscr{S}_{\hat{q}}(\mathfrak{S})$ of a surface $\mathfrak{S}$ is a quantization of the ${\rm SL}_n$-character variety, and is spanned over $\mathbb{Z}[\hat{q}^{\pm 1}]$ by framed tangles in $\mathfrak{S} \times (-1,1)$. If $\hat{q}$ is evaluated at a root of unity $\hatω$ with the order of $\hatω^{4n^2}$ being $N$, then for $\hatη = \hatω^{N^2}$, the Frobenius homomorphism $Φ: \mathscr{S}_{\hatη}(\mathfrak{S}) \to \mathscr{S}_{\hatω}(\mathfrak{S})$ is a surface generalization of the well-known Frobenius homomorphism between quantum groups. We show that the image under $Φ$ of a framed oriented knot $α$ is given by threading along $α$ of the reduced power elementary polynomial, which is an ${\rm SL}_n$-analog of the Chebyshev polynomial $T_N$. This generalizes Bonahon and Wong's result for $n=2$, and confirms a conjecture of Bonahon and Higgins. Our proof uses representation theory of quantum groups and its skein theoretic interpretation, and does not require heavy computations. We also extend our result to marked 3-manifolds.

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arXiv2024

The Unicity Theorem and the center of the ${\rm SL}_3$-skein algebra

The ${\rm SL}_3$-skein algebra $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ of a punctured oriented surface $\mathfrak{S}$ is a quantum deformation of the coordinate algebra of the ${\rm SL}_3$-character variety of $\mathfrak{S}$. When $\bar{q}$ is a root of unity, we prove the Unicity Theorem for representations of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$, in particular the existence and uniqueness of a generic irreducible representation. Furthermore, we show that the center of $\mathscr{S}_{\bar{q}}(\frak{S})$ is generated by the peripheral skeins around punctures and the central elements contained in the image of the Frobenius homomorphism for $\mathscr{S}_{\bar{q}}(\frak{S})$, a surface generalization of Frobenius homomorphisms of quantum groups related to ${\rm SL}_3$. We compute the rank of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ over its center, hence the dimension of the generic irreducible representation.

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arXiv2024

A trilogy of mapping class group representations from three-dimensional quantum gravity

For a punctured surface $\mathfrak{S}$, the author and Scarinci (arXiv:2112.13329) have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold $\mathfrak{S} \times \mathbb{R}$ of constant sectional curvature $Λ\in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group ${\rm MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $Λ$. In this survey article, we review and elaborate on this result.

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arXiv2024

A quantization of moduli spaces of 3-dimensional gravity

We construct a quantization of the moduli space $\mathcal{GH}_Λ(S\times\mathbb{R})$ of maximal globally hyperbolic Lorentzian metrics on $S\times \mathbb{R}$ with constant sectional curvature $Λ$, for a punctured surface $S$. Although this moduli space is known to be symplectomorphic to the cotangent bundle of the Teichmüller space of $S$ independently of the value of $Λ$, we define geometrically natural classes of observables leading to $Λ$-dependent quantizations. Using special coordinate systems, we first view $\mathcal{GH}_Λ(S\times\mathbb{R})$ as the set of points of a cluster $\mathscr{X}$-variety valued in the ring of generalized complex numbers $\mathbb{R}_Λ= \mathbb{R}[\ell]/(\ell^2+Λ)$. We then develop an $\mathbb{R}_Λ$-version of the quantum theory for cluster $\mathscr{X}$-varieties by establishing $\mathbb{R}_Λ$-versions of the quantum dilogarithm function. As a consequence, we obtain three families of projective unitary representations of the mapping class group of $S$. For $Λ<0$ these representations recover those of Fock and Goncharov, while for $Λ\geq 0$ the representations are new.

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arXiv2024

Naturality of ${\rm SL}_3$ quantum trace maps for surfaces

Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed ${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent examples of cluster $\mathscr{X}$-varieties and higher Teichmüller spaces. In a previous paper of the author (arXiv:2011.14765), building on the works of others, the so-called ${\rm SL}_3$ quantum trace map is constructed for each triangulable punctured surface $\frak{S}$ and an ideal triangulation $Δ$ of $\frak{S}$, as a homomorphism from the stated ${\rm SL}_3$-skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster $\mathscr{X}$-chart for $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ associated to $Δ$. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the ${\rm SL}_3$ quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum ${\rm SL}_3$-${\rm PGL}_3$ duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.

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arXiv2023

Three-dimensional quantum gravity from the quantum pseudo-Kähler plane

A new canonical Hopf algebra called the quantum pseudo-Kähler plane is introduced. This quantum group can be viewed as a deformation quantization of the complex two-dimensional plane $\mathbb{C}^2$ with a pseudo-Kähler metric, or as a complexified version of the well-known quantum plane Hopf algebra. A natural class of nicely-behaved representations of the quantum pseudo-Kähler plane algebra is defined and studied, in the spirit of the previous joint work of the author and I. B. Frenkel. The tensor square of a unique irreducible representation decomposes into the direct integral of the irreducibles, and the unitary decomposition map is expressed by a special function called the modular double compact quantum dilogarithm, used in the recent joint work of the author and C. Scarinci on the quantization of 3d gravity for positive cosmological constant case. Then, from the associativity of the tensor cube, and from the maps between the left and the right duals, we construct unitary operators forming a new representation of Kashaev's group of transformations of dotted ideal triangulations of punctured surfaces, as an analog of Kashaev's quantum Teichmüller theory. The present work thus inspires one to look for a Kashaev-type quantization of 3d gravity for positive cosmological constant.

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arXiv2023

${\rm SL}_2$ quantum trace in quantum Teichmüller theory via writhe

Quantization of the Teichmüller space of a punctured Riemann surface $S$ is an approach to $3$-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop $γ$ in $S$ gives rise to a natural trace-of-monodromy function $\mathbb{I}(γ)$ on the Teichmüller space. For any ideal triangulation $Δ$ of $S$, this function $\mathbb{I}(γ)$ is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of $Δ$. An important problem was to construct a quantization of this function $\mathbb{I}(γ)$, namely to replace it by a noncommutative Laurent polynomial in the quantum variables. This problem, which is closely related to the framed protected spin characters in physics, has been solved by Allegretti and Kim using Bonahon and Wong's ${\rm SL}_2$ quantum trace for skein algebras, and by Gabella using Gaiotto, Moore and Neitzke's Seiberg-Witten curves, spectral networks, and writhe of links. We show that these two solutions to the quantization problem coincide. We enhance Gabella's solution and show that it is a twist of the Bonahon-Wong quantum trace.

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arXiv2022

${\rm SL}_3$-laminations as bases for ${\rm PGL}_3$ cluster varieties for surfaces

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.

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arXiv2022

Three dimensional construction of the Virasoro-Bott group

We present a three-dimensional geometric construction of the Virasoro-Bott group, which is a central extension of the group of diffeomorphisms of the circle. Our approach is analogous to the well-known construction of a central extension of the loop group by means of the Wess-Zumino topological term. In particular, the Virasoro-Bott group is realized as a quotient group of diffeomorphisms of the disc with special boundary conditions. We identify the Lie algebra corresponding to our group with the Virasoro algebra. We also show that for generalized boundary conditions the Virasoro algebra is extended to a semidirect product with the Heisenberg algebra. We discuss the relation between our construction, the Chern-Simons theory, and the three-dimensional gravity.

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arXiv2020

Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants

Quantization of the Teichmüller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichmüller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic-analytic theory of the Shale-Weil intertwiners for the Schrödinger representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster $\mathscr{X}$-varieties.

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arXiv2019

Laurent positivity of quantized canonical bases for quantum cluster varieties from surfaces

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety, whose positive real points recover the enhanced Teichmüller space of $S$. Their basis is enumerated by integral laminations on $S$, which are collections of closed curves in $S$ with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an ideal triangulation of $S$, each quantum basis element is a Laurent polynomial in the exponential of quantum shear coordinates for edges of the triangulation, with coefficients being Laurent polynomials in $q$ with integer coefficients. We show that these coefficients are Laurent polynomials in $q$ with positive integer coefficients. Our result was expected in a positivity conjecture for framed protected spin characters in physics and provides a rigorous proof of it, and may also lead to other positivity results, as well as categorification. A key step in our proof is to solve a purely topological and combinatorial ordering problem about an ideal triangulation and a closed curve on $S$. For this problem we introduce a certain graph on $S$, which is interesting in its own right.

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arXiv2019

Phase constants in the Fock-Goncharov quantum cluster varieties

A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized to unitary maps between the Hilbert spaces intertwining the representations. These unitary intertwiners are described using the quantum dilogarithm function $Φ^\hbar$. Algebraic relations among classical mutations are satisfied by the intertwiners up to complex constants. The present paper shows that these constants are $1$. So the mapping class group representations resulting from the Chekhov-Fock-Goncharov quantum Teichmüller theory are genuine, not projective. During the course, the hexagon and the octagon operator identities for $Φ^\hbar$ are derived.

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arXiv2018

Finite dimensional quantum Teichmüller space from the quantum torus at root of unity

Representation theory of the quantum torus Hopf algebra, when the parameter $q$ is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a `multiplicity module' tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map ${\bf T}$ between the multiplicity modules, called the 6j-symbols. We study the left and right dual representations, and correspondingly, the left and right representations on the ${\rm Hom}$ spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map ${\bf A}$ on a multiplicity module, encoding the permutation of the roles of the irreducible representations in the identification of the multiplicity module as the space of intertwiners between representations. We show that ${\bf T}$ and ${\bf A}$ satisfy certain consistency relations, forming a Kashaev-type quantization of the Teichmüller spaces of bordered Riemann surfaces. All constructions and proofs in the present work uses only plain representation theoretic language with the help of the notions of the left and the right dual and Hom representations, and therefore can be applied easily to other Hopf algebras for future works.

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arXiv2018

Comments on Exchange Graphs in Cluster Algebras

An important problem in the theory of cluster algebras is to compute the fundamental group of the exchange graph. A non-trivial closed loop in the exchange graph, for example, generates a non-trivial identity for the classical and quantum dilogarithm functions. An interesting conjecture, partly motivated by dilogarithm functions, is that this fundamental group is generated by closed loops of mutations involving only two of the cluster variables. We present examples and counterexamples for this naive conjecture, and then formulate a better version of the conjecture for acyclic seeds.

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arXiv2016

Quasiphantom categories on a family of surfaces isogenous to a higher product

We construct exceptional collections of line bundles of maximal length 4 on $S=(C \times D)/G$ which is a surface isogenous to a higher product with $p_g=q=0$ where $G=G(32,27)$ is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements.

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