arXiv subjects
Publications and source records attributed to R. M. Kashaev.
The construction of quantum knot invariants from solutions of the Yang--Baxter equation (R-matrices) is reviewed with the emphasis on a class of R-matrices admitting an interpretation in intrinsically three-dimensional terms.
A Delta-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. By considering two simplest examples coming from knot theory, we illustrate how can one associate a Delta-groupoid to an ideal triangulation of a three-manifold. We also describe in detail the rings associated with the Delta-groupoids of these examples.
A Delta-groupoid is an algebraic structure which axiomitizes the combinatorics of a truncated tetrahedron. It is shown that there are relations of Delta-groupoids to rings, group pairs, and (ideal) triangulations of three-manifolds. In particular, one can associate a Delta-groupoid to ideal triangulations of knot complements. It is also possible to define a homology theory of Delta-groupoids. The constructions are illustrated by examples coming from knot theory.
The relationship (both classical and quantum mechanical) between the discrete Liouville equation and Teichmüller theory is reviewed.
Based on the notion of a $Δ$-group(oid), ring-valued invariants of pairs of topological spaces can be defined in intrinsic topological terms.
The q-binomial formula in the limit q->1 is shown to be equivalent to the Rogers five term dilogarithm identity.
Let M be the moduli space of irreducible flat PSL(2,R) connections on a punctured surface of finite type with parabolic holonomies around punctures. By using a notion of admissibility of an ideal arc, M is covered by dense open subsets associated to ideal triangulations of the surface. A principal bundle over M is constructed which, when restricted to the Teichmuller component of M, is isomorphic to the decorated Teichmuller space of Penner. The construction gives a generalization to M of Penner's coordinates for the Teichmuller space.
It is shown that the N-th power of the light-cone evolution operator of 2N-periodic quantum discrete Liouville model can be identified with the Dehn twist operator in quantum Teichmuller theory.
The notion of a symmetrically factorizable Lie group is introduced. It is shown that each symmetrically factorizable Lie group is related to a set-theoretical solution of the pentagon equation. Each simple Lie group (after a certain Abelian extension) is symmetrically factorizable.
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.
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 Hermitean conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.
The volume conjecture, formulated recently by H. Murakami and J. Murakami, is proved for the case of torus knots.
Modified universal R-matrices, associated with the central extension (through the Drinfeld's double construction) of the quantum groups U_q(sl_n), are realized through an infinite dimensional spectral parameter dependent solution for the tetrahedron equation, provided a certain identity on $q$-exponentials holds true.
In the quantum Teichmuller theory, based on Penner coordinates, the mapping class groups of punctured surfaces are represented projectively. The case of a genus three surface with one puncture is worked out explicitly. The projective factor is calculated. It is given by the exponential of the Liouville central charge.
We describe a scheme of constructing classical integrable models in 2+1-dimensional discrete space-time, based on the functional tetrahedron equation - equation that makes manifest the symmetries of a model in local form. We construct a very general "block-matrix model" together with its algebro-geometric solutions, study its various particular cases, and also present a remarkably simple scheme of quantization for one of those cases.
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.
The tetrahedron equation in a special substitution is reduced to a pair of pentagon and one ten-term equations. Various examples of solutions are found. $O$-doubles of Novikov, which generalize the Heisenberg double of a Hopf algebra, provide a particular algebraic solution to the problem.
The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number $N$. By the analysis of particular examples it is argued that for a hyperbolic knot (link) the absolute value of this invariant grows exponentially at large $N$, the hyperbolic volume of the knot (link) complement being the growth rate.