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added the statement of $I_8$ appearing as the $O(\ell^6)$ higher curvature correction to 11d supergravity (here):
A systematic analysis of the possible supersymmetric higher curvature corrections of D=11 supergravity makes the $I_8$ appear as the higher curvature correction at order $\ell^6$, where $\ell$ is the Planck length in 11d (Souères-Tsimpis 17, Section 4).
At this order, the equation of motion for the supergravity C-field flux $G_4$ and its dual $G_7$ is (Souères-Tsimpis 17, (4.3))
$d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge G_4(\ell) + \ell^6 I_8 \,,$where the flux forms themselves appear in their higher order corrected form as power series in the Planck length
$G_4(\ell) \;=\; G_4 + \ell^6 G_4^{(1)} + \cdots$ $G_7(\ell) \;=\; G_7 + \ell^6 G_7^{(1)} + \cdots$I think what was
$d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R)$should be
$d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - 2\ell^3 \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R)$as the $\ell^3$ correction term looks like it should be, based on the text, $\ell^3 G_4(\ell) \wedge \tfrac{1}{2} p_1(R)$, not $\tfrac{1}{2}G_4(\ell)\wedge \tfrac{1}{2}p_1(R)$. Is this right?
Yeah, but it’s more complicated:
They write $tr(R^2)$ instead of $p_1$. The conversion factor for thee two should be $1/4\pi^2$, but of course it depends on conventions. Worse, when deriving the $\ell^6$-correction, they ignore the $\ell^3$-correction but then “absorb a numerical factor” in $\ell^6$ (below (4.3)). So at this point the two $\ell$-s in the two prefactors are not even the same anymore.
So what I did in the entry was that I took for the $\ell^3$-term the prefactor that I know from other reasons to be there. Then the $\ell^6$ term is correct for some rescaling of $\ell$. Of course there should really be a comment on that step in the entry.
And it would be good to sort out the actual relative factor between these two correction terms. If you are interested in digging into this, please let me know.
Sure, I’d be interested to have a look. Teaching is over, so have a bit more time now.
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