The quantum dilogarithm and representations of quantum cluster varieties
By V. Fock, A. Goncharov. 2007
232 citations recorded by the source corpus.
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 are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/760e56d9eaf36e241224227232b465886460f176
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Quantum Riemann Surfaces in Chern-Simons Theory (2011, methodology): “nd q := e~~ ; (6.13) or via an integral formula ~=2 = exp 1 4 Z R+i dx x e ipx sinh(ˇx)sinh(~x=(2i)) : (6.14) Properties of this remarkable function have been discussed in many places, including [75, 19, 76, 77, 78, 79, 80, 2]. It was central to the state integral model constructed by [2, 20, 73].”
- Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I (2010, background): “The operators Uτ2,τ1 describe the change of representation when passing from the quantum theory associated to triangulation τ1 to the one associated to τ2 [7, 20, 41,77].”
- Cluster algebras and derived categories (2012, unknown): “…[61] [62] [63] [10] . . . ; • discrete dynamical systems [27] [47] [74] [79] [84] [81] [91] . . . ; • higher Teichmüller spaces [35] [36] [37] [38] [39] . . . ; • combinatorics and in particular the study of polyhedra like the Stasheff associahedra [21] [22] [42] [43] [71] [90] [102] [104] [103]…”
- Spectral Networks (2012, unknown): “that these spaces admit a natural quantization [12, 14].”
- R-Twisting and 4d/2d Correspondences (2010, background): “In fact the dual torus algebra generated by the operators [51, 52]”
- General Argyres-Douglas theory (2012, background): “The generalization of lamination to the disk is described in [62] and the description of the corresponding expression for the Darboux coordinates should be possible once we know the cluster coordinate.”
- Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, background): “ i> (x ′ i ) a0i ·(1−1/y′ 0 ) −1 . Up to a change of sign these are cluster transformations. Namely, if we set Xi= −yi,X′ i = −y′ i,Ai= 1/xi,i∈Ithen our formulas become formulas (17) and (18) from [23] (in the notation from loc. cit). Remark 31 Let us recall the variety Nfrom Section 2.6 defined b”
- Representations of Algebras and Related Topics (2011, methodology): “1 of [18] (they write q for the indeterminate we denote by q).”
- Framed BPS states (2010, unknown): “Indeed, concrete operator interpretations of the Xγ were constructed in [30] and in [17, 66, 67].”
- Geometry of canonical bases and mirror symmetry (2013, background): “There are cluster volume forms A and X on the A and X spaces [21], see Sect.”
- Chern-Simons theory and S-duality (2011, background): “Following [18] and [36], the induced automorphism of Â~ factors as a composition of a linear symplectic transformation and conjugation by a quantum dilogarithm:”