I am trying to understand what is a Higgs bundle as defined in this paper by Gukov and Pei. They say it is a pair $(E, \Phi)$

• $E$ is a holomorphic principal $G^\mathbb{C}$ bundle

• $\Phi \in H^0(\Sigma, \mathrm{ad}(E) \otimes K )$

Can anyone explain to me a little about this sheaf $\mathrm{ad}(E) \otimes K$ ?

In light of comments, I've now learned $K$ is the canonical bundle of $\Sigma$. Can we just say $\Phi$ is a "matrix of $E$-valued 1-forms over $\Sigma$ transforming in the adjoint representations of $G$"?