Spectral description of non-commutative local systems on surfaces and non-commutative cluster varieties

TL;DR

This paper develops a non-commutative geometric framework for surfaces, introducing spectral, cluster Poisson, and A-varieties, and linking them to Stokes data, with applications to integrable systems and quantization.

Contribution

It generalizes classical surface and cluster structures to a non-commutative setting, introducing new stacks, spectral surfaces, and cluster varieties with Poisson and A-structures.

Findings

Non-commutative stacks of flat vector bundles are birationally equivalent to spectral surface moduli.

Non-commutative cluster Poisson and A-varieties carry canonical structures and are acted upon by the mapping class group.

Stacks of Stokes data admit cluster structures and can be quantized.

Abstract

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 (m-1)(m-2)/2 holes. Generalising this, we prove that non-commutative stacks X(m,S) of framed rank m flat R-vector bundles of on decorated surfaces S are birationally identified with the moduli spaces of flat line bundles on a spectral surface assigned to certain bipartite graphs on S. We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry canonical non-commutative Poisson structure. The space X(m,S) has a structure of a non-commutative cluster Poisson variety, equivariant under the action of the mapping class group. For bipartite graphs on a torus, we get the non-commutative dimer cluster integrable system. We define non-commutative cluster A-varieties related to bipartite ribbon graphs. They carry canonical non-commutative 2-form. The non-commutative stack A(m,S) of twisted decorated local systems on S carries a cluster A-variety structure, equivariant under the action of the mapping class group. The non-commutative cluster A-coordinates on the space A(m,S) are ratios of Gelfand-Retakh quasideterminants. In the case m=2 this recovers the Berenstein-Retakh non-commutative cluster algebras related to surfaces. We introduce stacks of admissible dg-sheaves, and use them to give an alternative proof of main results. We show that any stack of Stokes data is a stack of admissible dg-sheaves of certain type. Using this we prove that all stacks of framed Stokes data carry a cluster Poisson structure, equivariant under the wild mapping class group. Therefore they can be equivariantly quantized. Similar stacks of decorated Stokes data carry an equivariant cluster A-variety structure.