papers

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, 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.

For a punctured surface 𝔖, the author and Scarinci have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold 𝔖×ℝ of constant sectional curvature Λ∊{-1,0,1}. The invariance of this quantization under the action of the mapping class group MCG(𝔖) of 𝔖 yields families of unitary representations of MCG(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.

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 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, defined and 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.

We construct a deformation quantization of the moduli space 𝓖𝓗Λ(S×ℝ) of maximal globally hyperbolic Lorentzian metrics on S×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 𝓖𝓗Λ(S×ℝ) as the set of points of a cluster 𝒳-variety valued in the ring of generalized complex numbers ℝΛ=ℝ[ℓ]/(ℓ^2+Λ). We then develop an ℝΛ-version of the quantum theory for cluster 𝒳-varieties by establishing ℝΛ-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≥0 the representations are new.

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.

Fock-Goncharov's moduli spaces 𝒳PGL3,𝔖 of framed PGL3-local systems on punctured surfaces 𝔖 provide prominent examples of cluster 𝒳-varieties and higher Teichmüller spaces. In a previous paper of the author (arXiv:2011.14765), the so-called SL3 quantum trace map is constructed for each triangulable punctured surface 𝔖 and its ideal triangulation ∆, as a homomorphism from the stated SL3-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 𝒳-chart for 𝒳PGL3,𝔖 associated to ∆. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations, and show that the SL3 quantum trace maps are compatible under these quantum mutation maps. As a result, the quantum SL3-PGL3 duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.

In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of G-local systems on a punctured surface 𝔖 with boundary data, when G is a group of type A2, namely SL3 and PGL3. Based on Kuperberg's SL3-webs, we introduce the notion of SL3-laminations on 𝔖 defined as certain SL3-webs with integer weights. We introduce coordinate systems for SL3-laminations, and show that SL3-laminations satisfying a congruence property are geometric realizations of the tropical integer points of the cluster 𝓐-moduli space 𝓐SL3,𝔖. Per each such SL3-lamination, we construct a regular function on the cluster 𝒳-moduli space 𝒳PGL3,𝔖. We show that these functions form a basis of the ring of all regular functions. For a proof, we develop SL3 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 𝒳PGL3,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 SL2-PGL2 case.

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 𝒳-varieties.

Quantization of Teichmüller space of a punctured Riemann surface S is an approach to three 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 𝕀(γ) on the Teichmüller space. For any ideal triangulation Δ of S, this function 𝕀(γ) is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arc of Δ. An important problem was to construct a quantization of this function 𝕀(γ), namely to replace it by a non-commutative 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-Wong's SL2 quantum trace for skein algebras, and by Gabella using Gaiotto-Moore-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 Bonahon-Wong's quantum trace.

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed PGL2-local systems on a punctured surface S. The moduli space is birational to a cluster 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.

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 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 Hom spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map 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 T and 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.

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.

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 Φℏ. 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 Φℏ are derived.

We construct exceptional collections of line bundles of maximal length 4 on S=(C×D)/G which is a surface isogenous to a higher product with pg=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.

We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon and Wong.

We define new coordinates for Fock-Goncharov's higher Teichmüller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group G. Some additional data on the boundary leads to two closely related moduli spaces, the 𝒳-space and the 𝒜-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of G=PGLm and G=SLm, together with Poisson structures. We consider new coordinates for higher Teichmüller spaces given as ratios of the coordinates of the 𝒜-space for G=SLm, which are generalizations of Kashaev's ratio coordinates in the case m=2. Using Kashaev's quantization for m=2, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for m=3, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.

Quantization of universal Teichmüller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group T. This yields certain central extensions of T by ℤ, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension T̂Kash of T resulting from the Kashaev quantization, and show that it corresponds to 6 times the Euler class in H^2(T;ℤ). Meanwhile, the braided Ptolemy-Thompson groups T∗, T♯ of Funar-Kapoudjian are extensions of T by the infinite braid group B∞, and by abelianizing the kernel B∞ one constructs central extensions T∗ab, T♯ab of T by ℤ, which are of topological nature. We show T̂Kash≅T♯ab. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension T̂CF of T resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to 12 times the Euler class and that T̂CF≅T∗ab. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.

We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator arises from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also show that the quantum universal Teichmüller space is realized in the infinite tensor power of the canonical representation naturally indexed by rational numbers including infinity. This suggests a relation to the same index set in the classification of projective modules over the quantum torus, the unitary counterpart of the quantum plane, and points to a new quantization of the universal Teichmüller space.

In this article we give a self contained existence proof for J. Conway's sporadic simple group Co1 [4] using the second author's algorithm [14] constructing finite simple groups from irreducible subgroups of GLn(2). Here n = 11 and the irreducible subgroup is the Mathieu group M24. From the split extension E of M24 by a uniquely determined 11-dimensional GF(2)M24-module V we construct the centralizer H = CG(z) of a 2-central involution z of E in an unknown target group G. Then we prove that all the conditions of Algorithm 2.5 of [14] are satisfied. This allows us to construct a simple subgroup G of GL276(23) which we prove to be isomorphic with Conway's original sporadic simple group Co1 by means of a constructed faithful permutation representation of G and Soicher's presentation [16] of the original Conway group Co1.

In this article we construct an irreducible simple subgroup G = <q, y, t, w> of GL8671(13) from an irreducible subgroup T of GL11(2) isomorphic to Mathieu's simple group M24 by means of Algorithm 2.5 of [13]. We also use the first author's similar construction of Fischer's sporadic simple group G1 = Fi23 described in [11]. He starts from an irreducible subgroup T1 of GL11(2) contained in T which is isomorphic to M23. In [7] J. Hall and L. S. Soicher published a nice presentation of Fischer's original 3-transposition group Fi24 [6]. It is used here to show that G is isomorphic to the simple commutator subgroup Fi'24 of Fi24. We also determine a faithful permutation representation of G of degree 306936 with stabilizer G1 = <q, y, w> ≅ Fi23. It enabled MAGMA to calculate the character table of G automatically. Furthermore, we prove that G has two conjugacy classes of involutions z and u such that CG(u) = <q, y, t> ≅ 2Aut(Fi22). Moreover, we determine a presentation of H = CG(z) and a faithful permutation representation of degree 258048 for which we document a stabilizer.

In the first section of this senior thesis the author provides some new efficient algorithms for calculating with finite permutation groups. They cannot be found in the computer algebra system MAGMA, but they can be implemented there. For any finite group G with a given set of generators, the algorithms calculate generators of a fixed subgroup of G as short words in terms of original generators. Another new algorithm provides such a short word for a given element of G. In the later sections, the author gives a self-contained existence proof for Fischer's sporadic simple group Fi23 using G. Michler's Algorithm [11] constructing finite simple groups from irreducible subgroups of GLn(2). This sporadic group was originally discovered by B. Fischer in [6] by investigating 3-transposition groups, see also [5]. This thesis gives a representation theoretic and algorithmic existence proof for his group. The author constructs the three non-isomorphic extenstions Ei by the two 11-dimensional non-isomorphic simple modules of the Mathieu group M23 over F=GF(2). In two cases Michler's Algorithm fails. In the third case the author constructs the centralizer H=CG(z) of a 2-central involution z of Ei in any target simple group G. This allows the author to construct G inside GL782(17). Its four generating matrices, character table and representatives for conjugacy classes are computed. It follows that G and Fi23 have the same character table.

In this article we give self-contained existence proofs for the sporadic simple groups Co2 and Fi22 using the second author's algorithm [10] constructing finite simple groups from irreducible subgroups of GLn(2). These two sporadic groups were originally discovered by J. Conway [4] and B. Fischer [7], respectively, by means of completely different and unrelated methods. In this article n=10 and the irreducible subgroups are the Mathieu group M22 and its automorphism group Aut(M22). We construct their five non-isomorphic extensions Ei by the two 10-dimensional non-isomorphic simple modules of M22 and by the two 10-dimensional simple modules of A22 = Aut(M22) over F=GF(2). In two cases we construct the centralizer Hi = CGi(zi) of a 2-central involution zi of Ei in any target simple group Gi. Then we prove that all the conditions of Algorithm 7.4.8 of [11] are satisfied.
This allows us to construct G3 ≅ Co2 inside GL23(13) and G2 ≅ Fi22 inside GL78(13). We also calculate their character tables and presentations.