V. V. Fock, A. B. Goncharov
The main result is a construction, via the quantum dilogarithm, of certain intertwiner operators, which play the crucial role in the quantization of the cluster X-varieties and construction of the corresponding canonical representation.
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Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are pro…
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.
Cyclic quantum Teichmüller theory
本文提出循环量子双对数的显式构造，揭示其在量子簇代数和中心字符中的双重作用，并通过几何方法分解量子态空间，同时证明了与量子泰西穆勒空间局部表示的互换子关系。
We extend the notion of the quantization of the coefficients of the ordinary cluster algebras to the generalized cluster algebras by Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain generalizations of the ordinary quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y-seeds.
We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case …
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 P2, we calculate the integral kernel of the corresponding operators in terms of Faddeev's quantum dilogarithm. Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the state-integrals appearing in three-manifold topology, and we show th…
A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized to unitary maps between the Hilbert spaces intertwining the representations. These unitary intertwiners are described using the quantum dilogarithm function $Φ^\hbar$. Algebraic relations among classical mutations are satisfied by the intertwiners up to complex constants. The pre…
Motivated by mathematical structures which arise in string vacua and gauge theories with N=2 supersymmetry, we study the properties of certain generalized theta series which appear as Fourier coefficients of functions on a twisted torus. In Calabi-Yau string vacua, such theta series encode instanton corrections from $k$ Neveu-Schwarz five-branes. The theta series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich-Soibelman symplectomorphisms. This effectively provides a quantum version of these transformat…
A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an $n\times n$ matrix with integer entries, or as a quiver in special cases, together with $n$ formal variables. A mutation is a certain rule for transforming a seed into another seed; the new variables are related to the previous variables by some rational expressions. To each seed one attaches an $n$-dimensional torus, and by gluing the tori along the birational maps defined by the mutation formulas, one …
In early eighties, Belavin and Drinfeld showed that nonskewsymmetric classical r-matrices for simple Lie algebras are classified by combinatorial objects which are now called Belavin-Drinfeld triples. Later the second author of the present paper generalized this result to the case of dynamical r-matrices, and showed that they correspond to generalized Belavin-Drinfeld triples. The dynamical r-matrix corresponding a triple is given by a certain explicit formula, which works for any symmetrizable Kac-Moody algebra. In special cases, this formula gives the Felder and the Belavin r-matrix. In 1994…
We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of U_q(sl_2)) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R-matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the inte…
A quantization of moduli spaces of 3-dimensional gravity
针对带洞曲面上具常截面曲率的极大全局双曲洛伦兹度量模空间，通过特殊坐标系将其视为广义复数环上的簇X-变体，构建依赖曲率的量子化，得到映射类群的三类射影酉表示，含新表示。
Three-dimensional quantum gravity from the quantum pseudo-Kähler plane
本文通过引入量子伪Kähler平面这一新典范Hopf代数，研究其表示论与模双紧量子二对数，并构造Kashaev变换群的新表示，从而为正宇宙学常数下的三维引力量子化提供新途径。
Geometries of Quantum Field Theories
本书介绍3d-3d对应，以自下而上方式从量子场论基本问题出发，探讨三维N=2超对称理论与三维流形几何的联系，强调理论空间中显现的几何与代数结构。
In previous work of the second- and third-named authors with Linhui Shen, cluster theory was used to construct wavefunctions for branes in threespace and conjecturally relate them to open Gromov-Witten invariants. This was done by defining a quantum Lagrangian subvariety of a quantum cluster variety, and mutating a simple solution to the defining equations in a distinguished seed. In this paper, we extend the construction to incorporate the skein-theoretic approach to open Gromov-Witten theory of Ekholm-Shende. In particular, we define a skein-theoretic version of cluster theory, including the…
In the context of 4D/2D dualities, SH$^c$ algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of $\mathcal{N}=2$ supersymmetric gauge theories. In this paper, we rewrite the SH$^c$ algebra in terms of three holomorphic fields $D_0(z)$, $D_{\pm1}(z)$ with which the algebra and its epresentations are simplified. The instanton partition functions for arbitrary $\mathcal{N}=2$ super Yang-Mills theories with $A_n$ and $A^{(1)}_n$ type quiver diagrams are compactly expressed as a product of four building blocks: Gaiotto state, dil…
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