Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alexander Goncharov, Maxim Kontsevich

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.

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

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.

Paper Structure

This paper contains 135 sections, 85 theorems, 319 equations, 71 figures.

Table of Contents

  1. Introduction
  2. Spectral description of generic triples of flags
  3. 1. Generic pairs of flags.
  4. 2. Spectral description of the groupoid of generic triples of flags.
  5. 3. Reconstruction functor revisited.
  6. 4. Spectral description of decorated triples of flags.
  7. 5. A generalization for surfaces glued from triangles.
  8. 6. The non-commutative Laurent phenomenon.
  9. Describing noncommutative local systems on surfaces
  10. 1. Spectral description of local systems on decorated surfaces via ${\rm GL_m}-$graphs.
  11. 2. Non-commutative cluster Poisson varieties.
  12. 3. An application: Non-commutative cluster integrable systems.
  13. 4. Spectral description of twisted local systems on decorated surfaces.
  14. 5. Non-commutative cluster ${\cal A}-$varieties.
  15. 6. Spectral description of the non-commutative decorated local systems on ${\Bbb S}$.
  16. ...and 120 more sections

Key Result

Lemma 1.1

Assigning to a generic pair of flags $({\cal A}, {\cal B})$ in an $m$-dimensional $R$-vector space $V$ the ordered collection of $m$ lines $({\rm L}^1, \ldots , {\rm L}^{m})$ we get an equivalence of groupoids:

Figures (71)

  • Figure 1: The bipartite graphs $\Gamma_2$ and $\Gamma_3$, shown by solid lines.
  • Figure 2: A bipartite graph $\Gamma_4$ for a generic triple of flags $({\cal A}, {\cal B}, {\cal C})$ in a $4$-dimensional space.
  • Figure 3: Vertices of the graph $\Gamma_3$ on the distance $\leq 2$ from the vertex ${\rm A}$ are on the two (purple) zig-zags.
  • Figure 4: The bipartite graph $\overline \Gamma_4$ includes the $\bullet-$vertices on the sides of the triangle.
  • Figure 5: There are four ${\rm A}-$paths from the vertex $v_3$ on the side $AB$ to the side $AC$.
  • ...and 66 more figures

Theorems & Definitions (175)

  • Lemma 1.1
  • proof
  • Theorem 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • Theorem 1.6
  • ...and 165 more