Are emotions a function of muscle contractions? If so, how does the map look like? I hope this is the right (or rightest) place to ask this question, I wasn't sure whether this question is more appropriate for cogsci or biology. After all, it does concern the biology of emotions.

To my knowledge, it is established research that the brain 'knows' what emotions you are having by examining their manifestations in the body. This absurd rube-goldbergian wiring can for instance be used to force oneself to become happier (at least temporarily) by smiling for a few minutes.

This brings up the question, whether emotions have any identity separate from a cluster of bodily sensations (or even more specifically, muscle contractions) at all.

It is certainly true (by introspection for e.g.) that 'negative' emotions are accompanied by muscle contradictions. On the other hand, there are several different 'negative' emotions that can be easily distinguished by the person having the emotion (e.g. shame and fear). Unfortunately I am not able to simply recall the sensations I had when I was ashamed or fearful from memory and compare them, but I would assume you could even tell from the outside, simply by looking at the person afflicted, which of the two emotions they are going through.

So, what does the research say? Are emotions uniquely recognizable by facial expression/muscle contractions? Or do emotions have some kind of 'hidden variables' in the brain?
Your brain mentally represents afferent bodily input as affect, or feelings of pleasantness/unpleasantness and arousal (Craig, 2002).  In certain theories (e.g., Barrett, 2017), your emotions are cognitive interpretations (conceptualizations) of your affect in a context.

In general, there is very weak evidence that discrete emotions correspond to specific and reliable bodily responses (for the most up-to-date meta-analysis, see Siegel et al., in press). 

There is also indeterminate evidence for specific and reliable correspondences with facial movements.  This has not been studied well enough to be known, although there is compelling evidence to suggest that there are no specific/reliable links on average (see Hillel Aviezer's work).

In general, emotion research is converging on the view that emotional events are highly variable, both within and across emotion categories, as well as within and across individuals and cultures. We see this variability in neural activity, physiology, behavior, hormones, vocalizations, facial movements, and so on.  From this perspective, there are no unique emotion signatures.