The quantum dilogarithm and representations of quantum cluster varieties

Integrability on the master space

“…This relation tell us that we are describing a g +1-dimensional Poisson manifold (there are g+2 variables, but one constraint among the faces) with d−1 Casimir operators (d−1 are the independent relation among the differences of d external perfect matchings) and 2I phase space variables. There are I commuting Hamiltonians and the system is classically (and quantum [6,7]) integrable.…”

“…It has indeed been observed that starting from the bipartite graph for the SCFT we have just described, one can construct a completely integrable system, in which the oriented loops of this graph are the dynamical variables. The Poisson manifold [7] associated to this system is the collection of bipartite diagrams (seeds) glued via Poisson cluster transformations. These transformations, also known as mutations, have been studied on quiver diagrams in [8].…”

Laminations from the symplectic double

“…In this case, the set C consists of the curves labeled c 1 and c 3 , while the set C • consists of the curves labeled c 2 and c 4 . In Sect.…”

Quantum deformation of planar amplitudes

“…Our considerations are based on consideration of quantum cluster algebras in the sense of [6] and of quantum spaces in the sense of [9,12]. However, we do not need the full strength of the theory of cluster algebras and our exposition does not depend on this theory.…”

“…By definition, a family of based tori T i and their skew fields of fractions F i is a poor man quantum X-space if every two fields of fractions in this family are connected by a sequence of mutations. The definitions above are downgraded versions of the definitions in [9,12]: we singled out only the properties of quantum spaces that are necessary for our proofs. Fock and Goncharov gave a construction of quantum spaces starting with some algebraic data.…”