freedom in four dimensions since the classical equations of motion force it to be constant. At the quantum level gauge fixing of the $3$-form gives two $2$-form ghosts with fermi statistics, three $1$-form ghosts with bose statistics, and four scalar ghosts with fermion statistics. This counting gives $ 4 - 2\cdot 6 + 3\cdot 4 - 4\cdot 1=0$ net components and so no propagating degrees of freedom, as expected. However, as we repeatedly stress, $2$-forms must be handled with care at the quantum level and that applies also to the two ghosts that accompany the $3$-form tensor. At the quantum level one $3$-form tensor contributes with $(-2)$ $2$-forms that cannot be naively dualized to scalars.

In summary, the duality frame that arises naturally through the AdS$_4$ compactification of 11D supergravity on $S^7$ gives a net of five antisymmetric 2-tensors: $$1 + 6 - 2= 5~.$$ In this duality frame the trace of the energy momentum tensor vanishes and there are no divergences [@Duff:1980qv].

This result does not invalidate our claim that there are divergences in ${\cal N}=8$ supergravity. On the contrary, it implicitly confirms the notion that different duality frames are quantum inequivalent since otherwise the distinction between $2$-forms and 