In case anyone else is having similar issues on Apple Silicon: I have found a workaround solution.
Currently, it seems that the file xAct/xPerm/mathlink/xperm.mac is not compatible with Apple Silicon chips.
This can be seen by starting Mathematica via the Terminal and trying to start xAct: one obtains the error messagesh: ~/Library/Mathematica/Applications/xAct//xPerm/mathlink/xperm.mac: Bad CPU type in executable.
The program can thus be executed only via Rosetta 2, which can be installed via
softwareupdate --install-rosetta
With this, xAct also runs on the new Apple Silicon chips via emulation, for the time being.
Are there perhaps plans to compile a new version which runs natively?
With best regards,
Thomas