In the paper "More On Gauge Theory And Geometric Langlands" by Edward Witten ( https://arxiv.org/abs/1506.04293 ), he writes about the Hitchin fibration
$\pi: \mathcal{M}_H \rightarrow \mathcal{V}, (E, \varphi) \mapsto Trace(\varphi^2)$
als the "fiber $\mathcal{F}$ of the hitchin fibration".
Now I would intepret "fiber" as $\pi^{-1}(p) \subset \mathcal{M}_H$ for some $ p \in \mathcal{V}$.
But on page 10 he states that "choosing a fiber means making a particular choice of $trace(\varphi^2)$. That's what I don't understand, since $\pi$ doesn't have to be injective and therefore there could be many choices of $\varphi$.
I understand that this question is hard to answer if you don't know the paper and it is asked a lot to read this paper just to answer my question but it would already help a lot if someone could tell me another interoretation of fiber here.
Thanks in advance!