I'm reading through appendix A of https://arxiv.org/abs/1311.4895 and I'm very confused with the diagram below in addition to the following (eqn (A3)). It says We consider once more the example triangulation shown in Fig. 16(d), we have
$$\Xi(a, b, c, d)=\Delta_{2}(a, b, c)+\Delta_{2}(c, b, d)$$
My confusion is this: Shouldn't I expect this equation to visually be represented as
Why is the plaquette instead drawn with only vertically up arrows and only horizontally rightward arrows? Am I incorrect that the simplex from b-c and the simplex from c-b cancel? Any help would be greatly appreciated!
The short answer to your question about your picture is "yes, you should expect the equation to visually be represented as ..."
On p. 16, the authors say "We extend this notation to include a direction." The choice of direction (a.k.a. orientation) doesn't matter, but you need to make that choice, and you should make a choice once for each simplex. For example, they have drawn an arrow from $b$ to $c$, thus defining $\Delta_1(b,c)$; this is also equal to $-\Delta_1(c,b)$.
Later they say that plaquette is defined by $$ \Xi(a,b,c,d) = \Delta_2(a,b,c) + \Delta_2(c,b,d) $$ and note the last term: $\Delta_2(c,b,d)$ has the edges oriented the same as in your picture, and the opposite of theirs. You could instead write $$ \Xi(a,b,c,d) = \Delta_2(a,b,c) - \Delta_2(b,c,d) $$ if you wanted to use the 2-simplices as oriented in their picture.