Rinat Kashaev

Quantization of Teichmüller Spaces and the Quantum Dilogarithm

The Teichmüller space of punctured surfaces with the Weil–Petersson symplectic structure and the action of the mapping class group is realized as the Hamiltonian reduction of a finite-dimensional symplectic space where the mapping class group acts by symplectic rational transformations. Upon quantization, the corresponding (projective) representation of the mapping class group is generated by the quantum dilogarithms.

Strongly Coupled Quantum Discrete Liouville Theory. I: Algebraic Approach and Duality

Abstract: The quantum discrete Liouville model in the strongly coupled regime, 1 < c < 25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is self-dual: there are two exponential fields related by Hermitian conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.

A LINK INVARIANT FROM QUANTUM DILOGARITHM

The link invariant, arising from the cyclic quantum dilogarithm via the particular R- matrix construction is proved to coincide with the invariant of triangulated links in S3 introduced in Ref. 14. The obtained invariant, like Alexander-Conway polynomial, vanishes on disjoint union of links. The R-matrix can be considered as the cyclic analog of the universal R-matrix associated with Uq(sl(2)) algebra.

(ZN×)n−1 generalization of the chiral Potts model

We show that theR-matrix which intertwines twon-by-Nn−1 state cyclicL-operators related with a generalization ofUq(sl(n)) algebra can be considered as a Boltzmann weight of four-spin box for a lattice model with two-spin interaction just as theR-matrix of the checkerboard chiral Potts model. The rapidity variables lie on the algebraic curve of the genusg=N2(n−1)((n−1)N-n)+1 defined by 2n–3 independent moduli. This curve is a natural generalization of the curve which appeared in the chiral Potts model. Factorization properties of theL-operator and its connection to the SOS models are also discussed.

On discrete three-dimensional equations associated with the local Yang-Baxter relation

The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to three dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable three-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov model is related to the star-triangle relation in the Ising model. In this Letter the corresponding discrete equation is derived. In the continuous limit it leads to a differential three-dimensional equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy.Some related operator solutions to the tetrahedron equation are also constructed.

QUANTUM DILOGARITHM AS A 6j-SYMBOL

The cyclic quantum dilogarithm is interpreted as a cyclic 6j-symbol of the Weyl algebra, considered as a Borel subalgebra BUq(sl(2)). Using modified 6j-symbols, an invariant of triangulated links in triangulated three-manifolds is constructed. Apparently, it is an ambient isotopy invariant of links.

A TQFT from Quantum Teichmüller Theory

By using quantum Teichmüller theory, we construct a one parameter family of TQFTs on the categroid of admissible leveled shaped 3-manifolds.

ON THE SPECTRUM OF DEHN TWISTS IN QUANTUM TEICHMÜLLER THEORY

The operator realizing a Dehn twist in quantum Teichmuller theory is diagonalized and continuous spectrum is obtained. This result is in agreement with the expected spectrum of conformal weights in quantum Liouville theory at c>1. The completeness condition of the eigenvectors includes the integration measure which appeared in the representation theoretic approach to quantum Liouville theory by Ponsot and Teschner. The underlying quantum group structure is also revealed.

Operators from Mirror Curves and the Quantum Dilogarithm

Mirror manifolds to toric Calabi–Yau threefolds are encoded in algebraic curves. The quantization of these curves leads naturally to quantum-mechanical operators on the real line. We show that, for a large number of local del Pezzo Calabi–Yau threefolds, these operators are of trace class. In some simple geometries, like local