Spectral Description of Non-Commutative Local Systems on Surfaces and Non-Commutative Cluster Varieties

Let R be a non-commutative field. We prove that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with \(\frac {1}{2}(m-1)(m-2)\) holes. Generalizing this, we prove that the non-commutative stack \(\mathcal {X}_{m, {\mathbb S}}\) of framed flat R-vector bundles of rank m on a decorated surface \({\mathbb S}\) contains open dense substacks, identified with stacks of flat line bundles on certain bipartite graphs \(\Gamma \) on \({\mathbb S}\) . We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry a canonical non-commutative Poisson structure. The result above just means that the space \(\mathcal {X}_{m, {\mathbb S}}\) has a structure of a non-commutative cluster Poisson variety, equivariant under the action of the mapping class group of \({\mathbb S}\) . For bipartite graphs on a torus, we get the non-commutative dimer cluster integrable system. We develop a parallel dual story of non-commutative cluster \(\mathcal {A}\) -varieties related to bipartite ribbon graphs. They carry a canonical non-commutative 2-form. The dual non-commutative moduli space \(\mathcal {A}_{m, {\mathbb S}}\) of twisted decorated local systems on \({\mathbb S}\) carries a cluster \(\mathcal {A}\) -variety structure, equivariant under the action of the mapping class group of \({\mathbb S}\) . The non-commutative cluster \(\mathcal {A}\) -coordinates on the space \(\mathcal {A}_{m, {\mathbb S}}\) are expressed as ratios of Gelfand-Retakh quasideterminants. In the case \(m=2\) this recovers the Berenstein-Retakh non-commutative cluster algebras related to surfaces. For any split reductive group \(\mathrm {G}\) with connected center, we prove that all stacks of framed \(\mathrm {G}\) -Stokes data carry a cluster Poisson structure, equivariant under the wild mapping class group. Therefore, these stacks can be equivariantly quantized. The similar stacks of decorated \(\mathrm {G}\) -Stokes data carry an equivariant cluster \(\mathcal {A}\) -variety structure. We introduce admissible dg-sheaves and define non-commutative stacks of Stokes data as stacks of admissible dg-sheaves of certain type.