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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 $G$"?

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    $\begingroup$ I haven't looked at that paper, but: Given a holomorphic principal $H$-bundle $E$, say $H:=G^{\mathbb{C}}$ for a compact Lie group $G$, and a (complex) representation $\rho:H\to \mathrm{GL}(V)$, you can form the associated holomorphic vector bundle $E\times^{H}V:=(E\times V)/H$ where $H$ acts on $E\times V$ by $h.(p,v):=(p\cdot h^{-1},\rho(h)v)$. Then $\mathrm{ad}(E)$ is defined to be the vector bundle associated to the adjoint representation of $H$ on its Lie algebra, i.e. $V:=\mathfrak{h}$ and $\rho:=\mathrm{ad}$. Here, $K=K_\Sigma$ is the canonical line bundle of $\Sigma$. $\endgroup$
    – Qfwfq
    Commented Dec 16, 2016 at 22:07
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    $\begingroup$ @johnmangual exactly, it is a matrix valued (you can think about it as a Lie algebra valued) top-form which for Higgs bundles is a holomorphic 1-form. $\endgroup$
    – Marion
    Commented Sep 3, 2017 at 12:33

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