So far in this book, we have extensively studied planar on-shell diagrams. In section 4.4, we introduced two natural classes of operations: amalgamation, the operation that allows us to build up very complex diagrams from very simple ones; and mergers and square moves, which allow us to connect very distinct on-shell diagrams that nevertheless encode the same physical information.

In this section we turn to the very obvious question that arises when dealing with planar diagrams of any sort: what are the corresponding dual graphs? what do they mean? and how are the operations we have found realized in terms of them? Of course, being two-colored, on-shell diagrams carry more information than ordinary graphs, and whatever definition of a dual graph we introduce must encode this additional information. Luckily, the theory of dual graphs for bipartite planar graphs is both known and simple; in fact, the dual of a bipartite graph is a familiar object in the physics of N =1 supersymmetric gauge theories: it is a quiver diagram! Indeed, the connection between bipartite graphs and quiver gauge theories is already an active research area in the physics community and has led to beautiful constructions such as those described in [49–54]. Bipartite graphs are also intimately related to dimer models, with the recent mathematical work [41] particularly closely related to our discussion.

The ‘dual’ of an on-shell diagram

Recall that the dual of an ordinary planar graph (one without colored vertices) is obtained by drawing a vertex for each face, and connecting adjacent faces with edges. In our case, we have graphs on a disc, and so the faces of an on-shell diagram can be divided into two distinct classes: those in the interior of the graph, and those on the exterior (those adjacent to the boundary of the disc).

As mentioned above, the dual of a bipartite graph turns out to be none other than an oriented quiver diagram. Let us now describe how this dual “quiver” of a general bipartite graph on a disc is defined. Let Г denote a bipartite graph on a disc; we define a flag F of Г to be the combination of one vertex of Г with one edge connected to it.