I'm very much a novice in these subjects, but Geometry of Interaction and Geometric Complexity Theory seem to speak similar language and have vaguely similar goals. Am I not mistaken? Are there any links between them?
No one knows if there are connections between GoI and GCT.
It's quite plausible, since both are used to analyze complexity, and since GoI is formulated in terms of monoidal categories and GCT is based on representation theory. However, to my knowledge there aren't any researchers who understand both well enough to say for certain.