Webs and multiwebs for the symplectic group

TL;DR

This paper introduces $2n$-multiwebs on planar graphs, explores their relation to $ ext{Sp}(2n)$-webs, and generalizes Kasteleyn's theorem to $2n$-multiweb covers, with classifications on simple surfaces.

Contribution

It defines $2n$-multiwebs, relates them to symplectic webs, and extends Kasteleyn's theorem to these structures on planar graphs.

Findings

Pfaffian of a matrix equals sum of traces of $2n$-multiwebs.

Generalization of Kasteleyn's theorem to $2n$-multiweb covers.

Classification of reduced $4$-webs and $2n$-webs on simple surfaces.

Abstract

We define $2n$-multiwebs on planar graphs and discuss their relation with $\mathrm{Sp}(2n)$-webs. On a planar graph with a symplectic local system we define a matrix whose Pfaffian is the sum of traces of $2n$-multiwebs. As application we generalize Kasteleyn's theorem from dimer covers to $2n$-multiweb covers of planar graphs with $U(n)$ gauge group. For $\mathrm{Sp}(4)$ we relate Kuperberg's ``tetravalent vertex'' to the determinant, and classify reduced $4$-webs on some simple surfaces: the annulus, torus, and pair of pants. We likewise define, for $\mathrm{Sp}(2n)$ and $q=1$, a $2n$-valent vertex corresponding to the determinant, and classify reduced $2n$-webs on an annulus.