Definition of Higgs bundle

Question

I currently try to deal with the definition of a Higgs bundle:

The definition is: $(E, \varphi)$ is called a Higgs bundle, if

$E$ is a holomorphic vector bundle and

$\varphi$ is a holomorphic 1-form with values in $End(E)$, s.t. $\varphi \wedge \varphi=0$

Now I am not sure what holomorphic 1-form with values in $End(E)$. Is it a section of $T^{*}M \otimes End(E)$? Since the definition of a 1-form in general is that it is a section in $T^{*}M$.

Also, I am reading a paper where it says

'Because $\varphi$ takes values in the adjoint representation, we can think of it locally as an $n \times n$ matrix of holomorphic one-forms – which we can take to act on the fiber of $E$.'

I don't understand what 'takes values in the adjoint representation' means here and how it is related to the above definition.

Thanks in advance for any help!

Answer 1

Yes, unfortunately, the standard definition (like the one in the Wikipedia article and the linked AMS Notices article) is sloppy, although, one can get used to it.

1. A holomorphic 1-form $\phi$ (on a complex manifold $M$) with coefficients in a holomorphic vector bundle $V\to M$ is a holomorphic section of the vector bundle $T^{(1,0)*}M \otimes V$. (By abusing the terminology, one identifies $T^{(1,0)*}_xM$ and $T^*_xM$.)

Assuming that $M$ is $m$-dimensional and $V=End(E)$, the endomorphism bundle of a holomorphic vector bundle $E\to M$ which has finite rank $n$, one can write $\phi$ (in local holomorphic coordinates) as $$ \sum_{j=1}^m A_j(z) dz^j, $$ where each $A_j$ is a holomorphic map $U\to Mat_{n\times n}({\mathbb C})=End({\mathbb C}^n)$, and $U\subset {\mathbb C}^m$ is a suitable open subset. This is what it means for $\phi$ to be a matrix-valued holomorphic 1-form or a matrix of holomorphic 1-forms.

The words adjoint representation are irrelevant here.

2. The adjoint representation becomes important when you want to make sense of the equation $\phi\wedge \phi=0$. If $\phi$ were a scalar-valued 1-form, then the wedge product would be the anti-symmetrization of the tensor product of $(1,0)$-forms. The question is how to deal with a tensor-field which is a section of $T^{(1,0)*}M \otimes E$. What is used in the context of Higgs fields is the Lie bracket on $End({\mathbb C}^n)$: $$ [a,b]= ab-ba, a, b\in End({\mathbb C}^n). $$ The latter is the Lie algebra ${\mathfrak g}$ of $GL_n({\mathbb C})$ and the adjoint representation of this Lie algebra is the representation given by the Lie bracket: $$ a\mapsto [a, \cdot], a\in {\mathfrak g}. $$ The expression $\phi\wedge \phi$ is then the usual wedge-product on $T^{(1,0)*}M$ and the Lie bracket of the (matrix-valued) coefficients. In local coordinates, $$ A(z)dz^i \wedge B(z) dz^j= [A(z), B(z)]dz^i \wedge dz^j. $$ In other words, the equation $\phi\wedge \phi=0$ simply means that the matrices $A_i(z), A_j(z)$ commute (for all $z$ and all $i, j$): $$ [A_i(z), A_j(z)]= 0, 1\le i< j\le m, z\in U. $$ If $M$ is a Riemann surface then this condition is satisfies automatically. (Each matrix, of course, commutes with itself.)

Thus, the slang with coefficients in the adjoint representation means the above rule for defining the wedge product of sections of $T^{(1,0)*}M \otimes End(E)$.