Existence of perfect matchings for weakly consistent dimer models on surfaces with or without boundary

A dimer model is a quiver with faces embedded in a surface, and its perfect matchings - sets of arrows meeting every face exactly once - drive applications from noncommutative crepant resolutions and cluster categorifications to the master and mesonic moduli spaces of bipartite field theories. Berggren and Serhiyenko (Forum of Mathematics, Sigma, 2025) proved that every simply connected weakly consistent dimer model has a perfect matching, but exhibited a weakly consistent model on a once-punctured torus with none, so weak consistency alone does not guarantee existence once the surface is not simply connected. The open problem is to determine which combinatorial or topological conditions restore existence of perfect matchings for weakly consistent dimer models on compact surfaces, with or without boundary; the authors' flagship instance asks whether every weakly consistent dimer model with no digons (no two-arrow faces) has a perfect matching. A resolution charts which hypotheses restore perfect-matching existence and bounds where weak consistency can coincide with strong consistency (nondegeneracy, every arrow lying in some perfect matching), thereby delimiting the reach of perfect-matching-based technology - nondegeneracy gradings, digon reduction, and the cluster categorification of dimer algebras - beyond disks and tori.

Background

A dimer model is a quiver (directed graph) $Q$ together with a set of faces, each face a cyclic path of arrows, embedded in a compact oriented surface $S$ , possibly with boundary, so that every arrow lies on exactly two faces - one read clockwise, one counter-clockwise around $S$ - or on one face together with the boundary of $S$ . Dimer models originated in statistical mechanics as models of diatomic molecules adsorbed on a lattice; the associated dimer algebra $A_Q$ is the Jacobian algebra of the quiver with potential determined by the faces, that is, the path algebra of $Q$ modulo relations identifying, for each face, the two paths running along its boundary. On the torus, dimer models (there called brane tilings) encode quiver gauge theories, and their dimer algebras are noncommutative crepant resolutions (endomorphism rings of tilting bundles serving as algebraic substitutes for resolutions of singularities) of Gorenstein toric threefolds.

A perfect matching of a dimer model $Q$ is a set $\mathcal{M}$ of arrows such that every face of $Q$ contains exactly one arrow of $\mathcal{M}$ (a chosen arrow serves both of the faces it borders; on general surfaces these are sometimes called almost perfect matchings). Passing to the dual bipartite graph, whose vertices are the faces of $Q$ and whose edges correspond to arrows, perfect matchings become ordinary matchings covering all internal vertices, so existence can be attacked with Hall-type marriage theorems, including versions for infinite locally finite bipartite graphs. On the torus, perfect matchings (dimer configurations) generate the perfect-matching polygon governing the center of the dimer algebra, and in the physics of bipartite field theories - quiver gauge theories defined by bipartite graphs on general Riemann surfaces with or without boundary - they compute the master and mesonic moduli spaces via determinants of Kasteleyn matrices, a signed adjacency matrix whose determinants enumerate perfect matchings.

The torus and disk theories rest on consistency conditions. To each dimer model one associates a zigzag (strand) diagram: for each arrow, a strand segment crossing it transversally between its two adjacent faces, with the segments glued into strands. A model is consistent when this strand diagram contains no bad configurations (forbidden crossing and lens patterns); this is equivalent to path-consistency - in every homotopy class of paths between two vertices there is, up to the face relations, a unique minimal path - and to cancellativity of the dimer algebra (cancellation: if paths $\alpha\beta$ and $\alpha\gamma$ sharing the initial arrow $\alpha$ are equal in $A_Q$ , then so are $\beta$ and $\gamma$ , and symmetrically from the right). These equivalences are proved for dimer models on surfaces other than the sphere. Consistent torus models are nondegenerate, meaning every arrow lies in at least one perfect matching, and their dimer algebras are Noetherian (satisfying the ascending chain condition on ideals) and 3-Calabi-Yau (admitting a self-dual resolution by finitely generated projective bimodules of length 3); on the disk, the same conclusions hold for Postnikov diagrams (consistent strand diagrams in the disk, underlying the cluster structure of positroid subvarieties of Grassmannians), where perfect matchings control the cluster categorification.

Berggren and Serhiyenko extended this framework to general compact oriented surfaces, with or without boundary. Their weak consistency, for a model not on a sphere, is any of the three equivalent conditions above (path-consistency, absence of bad strand configurations, cancellativity), and a model is strongly consistent when it is weakly consistent and nondegenerate. They prove that the dimer algebra of a finite strongly consistent model is bimodule internally 3-Calabi-Yau with respect to its boundary idempotents, which - when the model is additionally Noetherian and boundary-finite (the quotient of the completed dimer algebra by the boundary idempotent is finite-dimensional) and has no 1-cycles or 2-cycles (a loop at a vertex, respectively a pair of antiparallel arrows) - implies that the Gorenstein-projective modules (modules admitting a doubly-infinite resolution by projective bimodules) over the boundary algebra categorify the cluster algebra (a commutative algebra generated by variables grouped into clusters that transform by binomial exchange relations) of the underlying quiver; on the disk this recovers Pressland's categorification of positroid cluster structures.

The theory opens a gap exactly where the surface stops being simply connected. Berggren-Serhiyenko's Theorem 6.7 shows that every (possibly infinite) simply connected weakly consistent dimer model has a perfect matching - in particular the universal cover of any weakly consistent model does - while their Example 6.8 gives a weakly consistent model on a torus with an open disk removed that has no perfect matching at all: any perfect matching would have to contain one arrow of the model's digon (a face consisting of exactly two arrows) and then one of two specific arrow pairs, which excludes every arrow of a corner face. Moreover, on annuli there are weakly consistent models that possess perfect matchings yet are degenerate, so some arrow lies in no perfect matching. Beyond disks and tori, weak consistency thus no longer implies nondegeneracy, and the perfect-matching-dependent machinery - the nondegeneracy grading, digon reduction, and the categorification pipeline - applies only under extra hypotheses whose scope is unknown.

Problem Statement

Determine which conditions, beyond simple connectedness, force a weakly consistent dimer model on a compact oriented surface, with or without boundary, to admit at least one perfect matching, that is, a set of arrows meeting every face in exactly one arrow. Since nondegeneracy presupposes existence, any answer also informs the stronger delineation of where weak consistency coincides with strong consistency (nondegeneracy: every arrow lying in some perfect matching), but the target here is existence of at least one perfect matching.

The question is posed by Berggren and Serhiyenko immediately after their Example 6.8 of a weakly consistent model on a once-punctured torus with no perfect matching: 'what sort of conditions may we impose on a weakly consistent dimer model to necessitate the existence of some perfect matching? In particular, does any weakly consistent dimer model with no digons have a perfect matching?'

A full resolution settles the flagship instance - proving that every weakly consistent dimer model with no digons (no faces bounded by exactly two arrows) has a perfect matching, or refuting this with an explicit weakly consistent, digon-free model admitting none - and, in the event of refutation, identifies which other combinatorial or topological conditions do guarantee perfect-matching existence (the source question restricts no kind of condition; illustratively these may concern the surface topology, the zigzag/strand data, or the configuration of faces), so that the resulting theorem covers weakly consistent models beyond the simply connected case of Theorem 6.7; note that the digon-free hypothesis and simple connectedness are incomparable (simply connected weakly consistent models may contain digons, and digon-free models need not be simply connected). A refutation by explicit counterexample is a decisive partial answer to the source question; a positive proof for the digon-free case simultaneously supplies a condition guaranteeing existence on every compact oriented surface, with or without boundary.

References

Jonah Berggren, Khrystyna Serhiyenko, Consistent dimer models on surfaces with boundary, Forum of Mathematics, Sigma 13 (2025), e114. DOI: 10.1017/fms.2025.18, URL: https://doi.org/10.1017/fms.2025.18 (arXiv:2310.02454)
Jonah Berggren, Khrystyna Serhiyenko, Boundary algebras of positroids, preprint (2024). arXiv:2404.02886, URL: https://arxiv.org/abs/2404.02886
Karin Baur, Charlie Beil, A generalization of cancellative dimer algebras to hyperbolic surfaces, Mathematische Zeitschrift 312 (2026), no. 4, article 121. DOI: 10.1007/s00209-026-04000-z, URL: https://doi.org/10.1007/s00209-026-04000-z (arXiv:2101.11512)
Sebastián Franco, Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes, Journal of High Energy Physics 2012 (2012), article 141. DOI: 10.1007/JHEP11(2012)141, URL: https://doi.org/10.1007/JHEP11(2012)141 (arXiv:1207.0807)
Matthew Pressland, Calabi-Yau properties of Postnikov diagrams, Forum of Mathematics, Sigma 10 (2022), e56. DOI: 10.1017/fms.2022.52, URL: https://doi.org/10.1017/fms.2022.52 (arXiv:1912.12475)
İlke Çanakçı, Alastair King, Matthew Pressland, Perfect matching modules, dimer partition functions and cluster characters, Advances in Mathematics 443 (2024), 109570. DOI: 10.1016/j.aim.2024.109570, URL: https://doi.org/10.1016/j.aim.2024.109570 (arXiv:2106.15924)
Charlie Beil, Noetherian criteria for dimer algebras, Journal of Algebra 585 (2021), 294-315. DOI: 10.1016/j.jalgebra.2021.05.012, URL: https://doi.org/10.1016/j.jalgebra.2021.05.012 (arXiv:1805.08047)