In the paper I am currently reading, there is the following phrase (first line on page 74 in this document) :
Let $\sigma \in \Omega^2(M)$ be constant.
What does this mean? To be honest, thinking of $\omega \colon M \to \Lambda(T^*M)$ as a constant map does not make any sense since $\sigma$ needs to be a section. Also, a differential form could be constant iff $d\sigma = 0$, that is, constant is a synonym for closed.
In the context of the paper you are reading, these are forms on the torus $T^n$ (something that you should have mentioned in your question), which is a Lie group. Then the natural interpretation of the expression "a constant form" is that it is a form invariant under the action of the Lie group on itself. In other words, if you lift the form to $R^n$, it has constant coefficients.