Quantum Dilogarithm Function Research Articles

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 1980s 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.

Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

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 $$\Phi ^\hbar $$ . 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 $$\Phi ^\hbar $$ are derived.

Generating functions for weighted Hurwitz numbers

Double Hurwitz numbers enumerating weighted n-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of Sn generated by transpositions are determined by an associated weight generating function. A uniquely determined 1-parameter family of 2D Toda τ-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov’s double Hurwitz numbers for which the ramification is simple at all but two specified branch points; the case of Belyi curves, with three branch points, two with specified profiles; the general case, with a specified number of branch points, two with fixed profiles, the rest constrained only by the genus; and the signed enumeration case, with sign determined by the parity of the number of branch points. Using the exponentiated quantum dilogarithm function as a weight generator, three new types of weighted enumerations are introduced. These determine quantum Hurwitz numbers depending on a deformation parameter q. By suitable interpretation of q, the statistical mechanics of quantum weighted branched covers may be related to that of Bosonic gases. The standard double Hurwitz numbers are recovered in the classical limit.

On the semi-classical limit of scalar products of the XXZ spin chain

We study the scalar products between Bethe states in the XXZ spin chain with anisotropy |Δ| > 1 in the semi-classical limit where the length of the spin chain and the number of magnons tend to infinity with their ratio kept finite and fixed. Our method is a natural yet non-trivial generalization of similar methods developed for the XXX spin chain. The final result can be written in a compact form as a contour integral in terms of Faddeev’s quantum dilogarithm function, which in the isotropic limit reduces to the classical dilogarithm function.

Braiding operator via quantum cluster algebra

We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the -operator for which hyperbolic octahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev -matrix up to a simple gauge-transformation.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’.

Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of GL(n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question.

Stability conditions and quantum dilogarithm identities for Dynkin quivers

We study the fundamental groups of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(ΓNQ) of the Calabi–Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected. As an application, we show that the Donaldson–Thomas type invariant associated to Q can be calculated as a quantum dilogarithm function on its exchange graph. In the case of D(ΓNQ), we show that the faithfulness of the Seidel–Thomas braid group action (which is known for Q of type A or N=2) implies the simply connectedness of the space of stability conditions.

The pentagon relation and incidence geometry

We define a map \documentclass[12pt]{minimal}\begin{document}$S:{\mathbb {D}}^2\times {\mathbb {D}}^2 \dashrightarrow {\mathbb {D}}^2\times {\mathbb {D}}^2$\end{document}S:D2×D2⤏D2×D2, where \documentclass[12pt]{minimal}\begin{document}${\mathbb {D}}$\end{document}D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev–Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure—the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

Representation of the quantum plane, its quantum double, and harmonic analysis on $$GL_q^+(2,\mathbb{R })$$

We give complete detail of the description of the GNS representation of the quantum plane \(\mathcal{A }\) and its dual \({\widehat{\mathcal{A }}}\) as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary \(W\) is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes’ construction of the multiplicative unitary \(\mathbf{W}_m\), we give the GNS description of the quantum double \(\mathcal{D }(\mathcal{A })\) which is equivalent to \(GL_q^+(2,\mathbb{R })\). Furthermore, we study the fundamental corepresentation \(T^{\lambda ,t}\) and its matrix coefficients, and show that it can be expressed by the \(b\)-hypergeometric function. We also study the regular corepresentation and representation induced by \(\mathbf{W}_m\) and prove that the space of \(L^2\) functions on the quantum double decomposes into the continuous series representation of \(U_{q\widetilde{q}}(\mathfrak{gl }(2,\mathbb{R }))\) with the quantum dilogarithm \(|S_b(Q+2i\alpha )|^2\) as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.

Quantum Teichmüller space from the quantum plane

We derive the quantum Teichm\"uller 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 Teichm\"uller space is realized in the infinite tensor power of the canonical representation naturally indexed by rational numbers including the 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 Teichm\"uller space.

Generalized volume conjecture and the [formula omitted]-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function

We introduce and study the partition function Z γ ( M ) for the cusped hyperbolic 3-manifold M . We construct formally this partition function based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in his studies of the modular double of the quantum group. Following Thurston and Neumann–Zagier, we deform a complete hyperbolic structure of M , and we define the partition function Z γ ( M u ) correspondingly. This function is shown to give the Neumann–Zagier potential function in the classical limit γ → 0 , and the A -polynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and a punctured torus bundle over the circle.

HYPERBOLIC STRUCTURE ARISING FROM A KNOT INVARIANT II: COMPLETENESS

We study the geometry of a knot invariant defined in terms of the quantum dilogarithm function. We show that a hyperbolic structure naturally arises in the classical limit of the invariant; the completeness conditions can also be identified with the saddle point equations by studying a (1,1)-tangle.

HYPERBOLIC STRUCTURE ARISING FROM A KNOT INVARIANT

We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a noncompact analog of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The three-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions in gluing polyhedra arise naturally in the classical limit as the saddle point equation of our invariant.

Quantum Integrable Model on (2+1)-D Lattice

We construct the quantum integrable dynamical system on a (2+1)-dimensional lattice. We consider the q -commuting operators on 2-D lattice as a discretization of the current algebra, and two evolution operators are explicitly defined in terms of the quantum dilogarithm function.

The \({\rbb Z}_{\mibs N}\) Symmetric Quantum Lattice Field Theory The Quantum Group Symmetry, the Yang–Baxter Equation, and the Integrals of Motion

We study the quantum group symmetry of the \({\bb Z}_{N}\) symmetric lattice field theory, which is viewed as a discretization of the \(\widehat{s\ell (N)}\) affine Toda field theory. We reveal the quantum group structure, and construct explicitly the quantum local integrals of motion by use of the quantum dilogarithm function.