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For completeness I was deriving the equations of motion for the Bosonic sector of maximal sugra.

The Action principle is ##S=\int \star R -\frac{1}{2}\star F_4\wedge F_4 + \frac{1}{6} F_4\wedge F_4\wedge A_3## with ##F_4 = dA_3##. The subscripts denote the degree of the forms, when I switch to index-notation these are omitted. The star is the hodge star operator.

I succeeded in deriving the EFE for this theory of gravity ##R_{MN}-\frac{1}{2}Rg_{MN} = \frac{1}{12}\left( F_{MPQR}F_N^{PQR} - \frac{1}{8}F^2g_{MN}\right)## with ##F^2 = F_{MNOP}F^{MNOP}##.

When I vary this action with respect to the 3-form potential ##A_3## I get

[tex]d\star F_4 -\frac{1}{6}F_4\wedge F_4 = 0[/tex]

The problem I have here is that in this PDF, the coefficient of the CS-term is different.

Let me write out the relevant part of the variation below

[tex]\delta S = -\frac{1}{2} \int \star d(\delta A_3)\wedge F_4 + \star F_4\wedge d(\delta A_3) + \frac{1}{6} \int d(\delta A_3)\wedge F_4 \wedge A_3 + F_4\wedge d(\delta A_3)\wedge A_3 + F_4\wedge F_4 \wedge \delta A_3[/tex]

The first integral simplifies to ##-\int \star F_4 \wedge d(\delta A_3)## due to the identity ##\star A_p \wedge B_p = \star B_p\wedge A_p##.

Partial integration immediately gives the first term of my result (positive sign).

For the next integral I use that ##d(\delta A_3)\wedge F_4 \wedge A_3 = (-1)^{4\cdot 4} F_4 \wedge d(\delta A_3) \wedge A_3## which show the first 2 terms are the same. Moving the factor ##d(\delta A_3)## once more doesn't change anything.

After partial integration the second integral becomes ##\frac{1}{6}\int -2 d(F_4 \wedge A_3) \wedge \delta A_3 + F_4 \wedge F_4 \wedge \delta A_3##.

I figured I could exploit the fact that the Leibniz rule is modified, but because ##F_4## has degree 4 we don't pick up an extra sign. ##d(F_4\wedge A_3 = d(F_4)\wedge A_3 + (-1)^4 F_4\wedge dA_3 = 0+F_4\wedge F_4##

In conclusion I get an apparent discrepancy with the resource I found.

I haven't been able to confirm this resource though, the resources I found are mostly concerned with the transformation laws of the fields in the action (e.g. Supergravity by Freedman and Van Proeyen)

Am I making this big of a mistake?

Thanks,

Joris