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.


  1. The action is from "Supergravity in 11 dimensions" by CJS.

  2. The time-reversal argument is p6-7 of "Wick rotation and supersymmetry", and p6 of https://arxiv.org/abs/hep-th/0307152.


1 Answer 1


Ref. 1 and my Phys.SE answer here give an argument that only depends on the Wick rotation itself and standard definitions.

Concerning the title question the main point is that a differential form should be invariant under Wick rotation.

In Ref. 1 the bosonic part of the 11D SUGRA action is listed in eqs. (2.9+12) in Minkowski signature and in eqs. (2.26+27) in Euclidean signature. There is a helpful Table 1 on p. 5 that lists different sign conventions in the literature.


  1. A. Bilal & S. Metzger, arXiv:hep-th/0307152.

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