Modular quantum dilogarithm

introduction

• introduced by Ludvig Faddeev
• called the modular quantum dilogarithm or regularized quantum dilogarithm
• applications
  • S-matrix of the quantum sine-Gordon model
• definition \[b\in \mathbb{C}\] with non-zero real part and \(z\in \mathbb{C}\)

\[ \Phi_{b}(z)=\exp\left(-\frac{1}{4}\int \frac{e^{-2zx\sqrt{-1}}}{\sinh(xb) \sinh(x/b)}\, \frac{dx}{x}\right) \]

related items

• Yang-Baxter equation (YBE)

articles

• D. Chicherin, V. P. Spiridonov, The hyperbolic modular double and Yang-Baxter equation, arXiv:1511.00131 [math-ph], October 31 2015, http://arxiv.org/abs/1511.00131
• Ip, Ivan C. H. “On Tensor Product Decomposition of Positive Representations of \(\mathcal{U}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))\).” arXiv:1511.07970 [math], November 25, 2015. http://arxiv.org/abs/1511.07970.
• Chicherin, D., and V. P. Spiridonov. “The Hyperbolic Modular Double and Yang-Baxter Equation.” arXiv:1511.00131 [hep-Th, Physics:math-Ph], October 31, 2015. http://arxiv.org/abs/1511.00131.
• Chicherin, D., and S. Derkachov. ‘Matrix Factorization for Solutions of the Yang-Baxter Equation’. arXiv:1502.07923 [hep-Th, Physics:math-Ph], 27 February 2015. http://arxiv.org/abs/1502.07923.
• Garoufalidis, Stavros, and Rinat Kashaev. ‘Evaluation of State Integrals at Rational Points’. arXiv:1411.6062 [hep-Th], 21 November 2014. http://arxiv.org/abs/1411.6062.
• Faddeev, Ludwig D., and Leon A. Takhtajan. “On the Spectral Theory of One Functional-Difference Operator from Conformal Field Theory.” arXiv:1408.0307 [hep-Th, Physics:math-Ph], August 1, 2014. http://arxiv.org/abs/1408.0307.
• http://arxiv.org/abs/1305.4291
• Johnson, P. R. 1996. “Exact Quantum S-matrices for Solitons in Simply-laced Affine Toda Field Theories.” arXiv:hep-th/9611117 (November 15). doi:10.1016/S0550-3213(97)00239-3. http://arxiv.org/abs/hep-th/9611117.
  • https://docs.google.com/file/d/0B8XXo8Tve1cxUXc3WXNKbjEycHc/edit
• Faddeev, L. 1995. “Discrete Heisenberg-Weyl Group and Modular Group.” arXiv:hep-th/9504111 (April 21). doi:10.1007/BF01872779. http://arxiv.org/abs/hep-th/9504111.
• Faddeev, L. D. 1994. “Current-Like Variables in Massive and Massless Integrable Models.” arXiv:hep-th/9408041 (August 7). http://arxiv.org/abs/hep-th/9408041.