Resurgence of Faddeev's quantum dilogarithm

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.

消失与定向折叠锥形的Riemann-Hilbert问题

Crosscap resurgence and Riemann--Hilbert problems for orientifolded conifolds

本文研究了大N SO/Sp 折叠锥形自由能的crosscap部分，通过q-Pochhammer塔和Faddeev量子对数积分的函数方程，推导了Borel变换、极点结构和Stokes跃迁等性质，并结合解析总结定理得到大N定向折叠自由能的消失陈述。

Rational hypergeometric identities

A special singular limit $ω_1/ω_2\to 1$ is considered for the Faddeev modular quantum dilogarithm (hyperbolic gamma function) and the corresponding hyperbolic integrals. It brings a new class of hypergeometric identities associated with bilateral sums of Mellin-Barnes type integrals of particular Pochhammer symbol products.

The Yang-Baxter relation and gauge invariance

Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group $A$, we construct an operator solution of the Yang-Baxter equation generalizing the solution of the Faddeev-Volkov model. Based on a specific choice of a subgroup $B\subset A$ and by using the Weil transformation, we also give a new non-operator interpretation of the Yang-Baxter relation. That allows us to construct a lattice QFT-model of IRF-type with gauge invariance under independent $B$-translations of local `spin' variables.

The Endless Beta Integrals

We consider a special degeneration limit $ω_1\to - ω_2$ (or $b\to {\rm i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the ${\rm SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperb…

通过量子二重对数实现指数变量拓扑递归中的 x-y 对偶

$x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm

本文利用 Faddeev 量子二重对数修改 $n=1$ 部分，将拓扑递归的 $x-y$ 对偶公式从代数曲线推广到指数变量谱曲线，并应用于拓扑顶点曲线等 Gromov-Witten 不变量计算。

Heisenberg double and Drinfeld double of the quantum superplane

We study infinite dimensional generalisations of the Heisenberg doubles of the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$ and find associated canonical elements which satisfy pentagon equation. The former reproduces the canonical element, expressed using the Faddeev's quantum dilogarithm, which has been found by Kashaev to be realised within quantised Teichmüller theory, while for the latter we show that it corresponds to an operator from quantised super Teichmüller theory. We study infinite dimensional representations of those two Heisenberg doubles and, using an algebra homomorphism b…

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 1980's 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 phas…