The bosonic sector of the Cremmer-Julia-Sherk (CJS) 11D supergravity action is
$$ eR-\frac12 F\wedge *F-\frac16 A\wedge F\wedge F,$$
where $F=dA$ is a 4-form field strength.
How would one perform a Wick rotation on this action? I know the grav/kinetic terms swap signs in the Euclidean action, and the Chern-Simons term becomes imaginary, but I can't prove it. Heuristically, one might say it's from the epsilon symbol in the CS term indicating that the term is topological, but that's not convinving to me. Apparently one can make time-reversal arguments to fix the signs, but that seems hand-wavy when explicit formulae should exist for such coordinate changes/analytic continuation.