Q-Diagrams in Threefolds

• Q-diagrams in threefolds are finite collections of smooth, cooriented surfaces in a three-manifold that, through specified intersection conditions, form the basis for singular Lagrangian structures.
• They underpin a structure theory for singular Lagrangians in cotangent bundles, facilitating the study of moduli of local systems and the categorification of cluster varieties within both commutative and non-commutative frameworks.
• Concrete examples, such as cube configurations and ideal triangulations, illustrate how Q-diagrams encode cluster exchange relations and topological data, linking combinatorial geometry with symplectic and categorical structures.

A Q-diagram in a threefold is a geometric and combinatorial object defined as a finite collection of smooth cooriented surfaces within a three-manifold $M$, with specified intersection and local configuration properties. Q-diagrams serve as 3-dimensional analogs of bipartite ribbon graphs and underpin a structure theory for singular Lagrangians in cotangent bundles, which in turn relate to moduli of local systems and cluster-type Lagrangians both in commutative and non-commutative frameworks. These constructions are foundational for the study of K$_2$-Lagrangians, categorification of cluster varieties, and symplectic geometry of threefolds (Goncharov et al., 12 Jan 2026).

1. Definition and Structure of Q-Diagrams

A Q-diagram $\mathcal{Q}$ in a threefold $M$ is a finite collection of smooth, cooriented surfaces $\{S_i\}_{i\in I}$. Each $S_i$ carries a coorientation, equivalently an orientation when $M$ is oriented, determined by a connected component of its conormal bundle $T^*_{S_i}M\setminus 0$.

The defining conditions for a Q-diagram are:

1. Any intersection of fewer than three surfaces is a transverse codimension-$k$ submanifold of $M$ for $k<3$.
2. All remaining intersection points are isolated and realized as quadruple points, being the intersection of exactly four surfaces $S_{i_0},\dots,S_{i_3}$.
3. At each quadruple point $q$, the conormals $h_0,\dots,h_3\in T_q^*M$ obey a unique positive linear relation:

$$\alpha_0\,h_0+\alpha_1\,h_1+\alpha_2\,h_2+\alpha_3\,h_3=0\,,\quad \alpha_j>0$$

Shifting any one $S_{i_j}$ in the direction of its coorientation near $q$ causes the four sheets to bound a simplex (possibly singular tetrahedron) with cooriented outward faces. This is the "shifting to simplex" condition (Goncharov et al., 12 Jan 2026).

2. Associated Singular Lagrangians in Cotangent Bundles

To a Q-diagram $\mathcal{Q}$ in $M$ corresponds a singular Lagrangian $\mathbb{L}\subset T^*M$, constructed as

$$\mathbb{L} = M^0 \cup \bigsqcup_{i\in I} T^*_{S_i}M$$

where $M^0$ is the complement in $M$ of the union of all "mixed" domains. This $\mathbb{L}$ unites the zero section over $M^0$ and the conormal bundles to each $S_i$. Aside from quadruple points, $\mathbb{L}$ is locally the union of two transverse smooth Lagrangian sheets; at quadruple points, it exhibits an ordinary codimension-two singularity (a local model is the cone on $S^1\times S^1$). This structure generalizes Lagrangians arising from flat bundles extending over threefolds (Goncharov et al., 12 Jan 2026).

3. Boundary at Infinity and Associated Moduli Stacks

$\mathbb{L}$ admits a "collaring" at infinity, conceptually as cones $[0,\infty)\times S_i$. The boundary at infinity, denoted $\partial{\mathbb{L}}$, is constructed as follows:

• The symplectic manifold

$$\mathcal{S} = (T^*\partial M) \cup_{\partial S_i\times[0,1]} \bigsqcup_i T^*S_i$$

• The Lagrangian boundary

$$\partial{\mathbb{L}} = (\partial M \cup \bigsqcup_i T^*_{\partial S_i}\partial M) \cup_{\partial S_i\times[0,1]} \bigsqcup_i S_i \subset \mathcal{S}$$

To any conic Lagrangian $L\subset T^*M$ is attached the dg-stack $\mathcal{X}_L$ of admissible (microlocal rank-one) dg-sheaves on $M$ microsupported in $L$. The boundary Lagrangian $\partial L \subset \mathcal{S}$ admits a symplectic dg-stack $\mathcal{X}_{\partial L}$, and there is a restriction functor

$$\mathrm{Res}:\;\mathcal{X}_L\;\longrightarrow\;\mathcal{X}_{\partial L}$$

whose image forms a derived Lagrangian substack (Goncharov et al., 12 Jan 2026).

4. Cluster and K$_2$-Lagrangian Descriptions

When $M$ is a threefold with boundary and $\mathcal{Q}$ is a Q-diagram of discs ($S_i\cong D^2$) with an alternating arrangement of boundary loops, the restriction functor identifies $\mathcal{X}_{\partial\mathbb{L}}$ with the moduli space ${\rm Loc}_1(\Upsilon)$ of rank-one $R$-local systems on a closed surface $\Upsilon$ (obtained by gluing the $S_i$ to the spectral surface determined by the boundary configuration).

The Lagrangian image $$\mathcal{L}_{\mathcal{Q}}\subset{\rm Loc}_1(\Upsilon)$$ is described by cluster exchange relations. At a quadruple point $q$, local cluster variables $X_1, X_2, X_3$ satisfy:

$X_1X_2X_3 + X_2X_3 + X_3 = 0 \qquad X_1X_2X_3 = -1$

For $R$ commutative, this reduces to $X_2X_3+X_3=1$, capturing the non-commutative cluster exchange for the octahedral (2↔2-move) transformation. Thus, $\mathcal{L_Q}$ is a K$_2$-Lagrangian when $R$ is commutative (Goncharov et al., 12 Jan 2026).

5. Non-Commutative Generalization

The framework extends naturally to arbitrary (skew) fields $R$. In this context, "local systems" are twisted flat $R$-line bundles (monodromy $-1$ around fibers of cotangent circles), equivalently microlocal rank-one $R$-dg-sheaves with specific boundary trivializations. All cluster mutations and associated symplectic or K$_2$ forms are defined in a purely non-commutative setting. These non-commutative cluster varieties, their mutation theory, and cluster Lagrangians, have explicit algebraic and geometric descriptions, with categorical enhancements via dg-stacks (Goncharov et al., 12 Jan 2026).

6. Illustrative Constructions and Examples

Two prominent examples elucidate the key mechanisms of Q-diagrams in threefolds:

Example	Threefold $M$	Q-Diagram & Structure	Cluster Lagrangian/Relations
Cube	Unit cube	Four central planes via principal diagonals, cooriented	Basic cluster in a 5-torus $(R^\times)^5$; relation $XYZ+YZ+Z=0$, $-XYZ=1$
Ideal Triangulation	Any 3-manifold with ideal triangulation	For $m\geq2$, $4m$ discs by introducing half-integral planes in each tetrahedron	Singularity graph dual to hypersimplicial decomposition; cluster Lagrangian in moduli of framed $\mathrm{GL}_m$-local systems on $\partial M$

These explicit constructions illustrate the correspondence between combinatorics of Q-diagrams, cluster exchange structures, and moduli of local systems (Goncharov et al., 12 Jan 2026).

7. Relationship to Quiver 3-Folds and Higher Rank Categorifications

Q-diagrams admit a combinatorial quiver-theoretic interpretation; an ideal triangulation of a base surface $S$ lifts to a quiver $Q(\Delta_m)$ via its refinement, encoding the data of black and white triangles, their cyclic interrelation, and associated cluster-type potentials $W$. In threefolds fibred by $A_m$-surfaces, the data of $(Q,W)$ organizes the structure of Calabi–Yau 3-categories, which are quasi-isomorphic to subcategories of the sign-twisted Fukaya category of the total space.

Cluster-like exchange relations, as manifest in wall-crossing and mutation sequences, solidify the deep connection between the topology of Q-diagrams in threefolds and their categorical and symplectic avatars, leading to categorifications of cluster Poisson varieties of framed local systems. This paradigm realizes structures that encode BPS spectra, stability conditions, and deformation structures central to geometry and mathematical physics (Smith, 2020).