Differential graded algebras for trivalent plane graphs and their representations

Abstract

To any trivalent plane graph embedded in the sphere Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the Casals–Murphy dg-algebra to non-commutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank $r$ representations of this dg-algebra, over a field $\mathbb{F}$, correspond to colorings of the faces of the graph by elements of the Grassmannian $\mathrm{Gr}(r,2r;\mathbb{F})$ so that bordering faces are transverse, up to the natural action of ${\mathrm{PGL}}_{2r}(\mathbb{F})$. Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that, for Legendrian 2-weaves, rank $r$ representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank $r$. This is the first explicit such computation of the bijection between the moduli spaces of representations and sheaves for an infinite family of Legendrian surfaces.