The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory

Invariants of 3-manifolds via link polynomials and quantum groups

The generalization of Jones polynomial of links to the case of graphs inR
3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups.

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Ribbon graphs and their invariants derived from quantum groups

State sum invariants of 3 manifolds and quantum 6j symbols

We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra
 representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the “face” formulation for any type of Lie algebra
 
$$\mathfrak{g}$$

 and arbitrary finite-dimensional representations of
 . We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq→1 these solutions degenerate again into
 solutions with
 
$$q' = \exp \left( {\frac{{2\pi i}}{{k + g}}} \right)$$

 . We also study the simples examples of solutions of our holonomic difference equations associated to
 
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}(2)})$$

 and find their expressions in terms of basic (orq−)-hypergeometric series. In the special case of spin −1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.

/pdf/quantum-affine-algebras-and-holonomic-difference-equations-a21oed7v2n.pdf

Quantum affine algebras and holonomic difference equations

Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices Bibliography

Yangians and Classical Lie Algebras

Publisher Summary
This chapter focuses on the quantization of lie groups and lie algebras. The Algebraic Bethe Ansatz—the quantum inverse scattering method—emerges as a natural development of the various directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution, the quantum theory of magnets, the method of commuting transfer-matrices in classical statistical mechanics, and factorizable scattering theory. The chapter discusses quantum formal groups, a finite-dimensional example, an infinite-dimensional example, and reviews the deformation theory and quantum groups.

https://inspirehep.net/files/33b6ead3f196ec7bd10f771d606096d9

Quantization of Lie Groups and Lie Algebras

We present an explicit formula for the deformation quantization on K\"{a}hler manifolds.

Deformation Quantization of Kahler Manifolds

We study families of quantum field theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g-1, and for the case g=0 - by smooth closed Jordan curves on the complex plane. In both cases we define a notion of tau-function as a partition function of the theory and evaluate it explicitly. For the case g > 0 the tau-function is an analytic torsion [3], and for the case g=0 - the regularized energy of a certain natural pseudo-measure on the interior domain of a closed curve. For these cases we rigorously prove the Ward identities for the current correlation functions and determine them explicitly. For the case g > 0 these functions coincide with those obtained in [21,36] using bosonization. For the case g=0 the tau-function we have defined coincides with the tau-function introduced in [29,44,24] as a dispersionless limit of the Sato's tau-function for the two-dimensional Toda hierarchy. As a corollary of the Ward identities, we obtain recent results [44,24] on relations between conformal maps of exterior domains and tau-functions. For this case we also define a Hermitian metric on the space of all contours of given area. As another corollary of the Ward identities we prove that the introduced metric is Kahler and the logarithm of the tau-function is its Kahler potential.

L. A. Takhtajan

Papers

Quantization of Lie Groups and Lie Algebras

Quantization of Lie Groups and Lie Algebras

Deformation Quantization of Kahler Manifolds

Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves I. Current correlation functions