Non-commutative cluster Lagrangians

TL;DR

This work introduces Q-diagrams—3d analogues of bipartite ribbon graphs—to define Lagrangian structures in cotangent spaces and boundary spectral data for threefolds. It develops a non-commutative cluster framework where stacks of microlocal dg-sheaves with rank-one microlocal support yield cluster Poisson varieties on surfaces and cluster Lagrangians in threefolds, controlled by Q-diagrams and their admissible deformations. A central achievement is the cube example, where a basic ${oldsymbol{b Q}}$-diagram is described by explicit equations in A- and Loc-coordinates, exposing an ${ m A}_4$-symmetry and linking to two-by-two mutations. The paper further connects to non-commutative character varieties, proposing hypersimplicial decompositions and spectral covers as tools to describe Lagrangian boundaries and their restrictions, with broad applications to moduli of flat local systems and their cluster structures. Overall, the framework provides a concrete, geometric, and algebraic route to non-commutative cluster Lagrangians tied to 3d diagrammatic combinatorics and their boundary data.

Abstract

The space Loc(m,S) of rank m flat bundles on a closed surface S is K_2-symplectic. A threefold M bounding S gives rise a K_2-Lagrangian in Loc(m,S) given by the flat bundles on S extending to M. We generalize this, replacing the zero section in the cotangent bundle to M by certain singular Lagrangians. First, we introduce Q-diagrams in threefolds. They are collections Q of smooth cooriented surfaces, intersecting transversally everywhere but in a finite set of quadruple crossing points. We require that shifting any surface of the collection from such a point in the direction of its coorientation creates a simplex with the cooriented out faces. The Q-diagrams are 3d analogs of bipartite ribbon graphs. Let L be the Lagrangian in the cotangent bundle to M given by the union of the zero section and the conormal bundles to the cooriented surfaces of Q. Let X(L) be the stack of admissible dg-sheaves on M with the microlocal support in L, whose microlocalization at the conormal bundle to each cooriented surface of Q is a rank one local system. We introduce the boundary dL of L. It is a singular Lagrangian in a symplectic space, providing a symplectic stack X(dL), and a restriction functor from X(L) to X(dL). The image of the latter is Lagrangian. We show that, under mild conditions on Q, this Lagrangian has a cluster description, and so it is a K_2-Lagrangian. It also has a simple description in the non-commutative setting.

Figures (36)

• Figure 1: ${\cal Q}-$diagrams of oriented curves on a surface are triple crossing diagrams.
• Figure 2: Alternating diagrams of points on a line, and of oriented curves on a surface.
• Figure 3: (Co)oriented lines and $\bullet$, $\circ$ and mixed domains for an alternating diagram.
• Figure 4: Resolving triple crossings we get a bipartite graph.
• Figure 5: The surface $\Upsilon$ is obtained by gluing surfaces $S_i$ to the spectral surface $\Sigma_{\cal T}$ along the loops $\alpha_{i*}$.

Theorems & Definitions (89)

• Definition 1.1
• Lemma 1.2
• Lemma 1.3
• Definition 1.4
• Lemma 1.5
• Lemma 1.6
• Definition 1.7
• Definition 1.8
• Lemma 1.9
• proof