3-form with components $a_{IJK}$ is reduced into various lower forms in 4D including 3-forms and 2-forms, $$\begin{aligned}
a_{\mu\nu\sigma}(x,y) &=b_{\mu\nu\sigma}(x) Y(y)~, \\ \nonumber
a_{\mu \nu p}(x,y) &=b_{\mu \nu}(x) Y_p^{(CE)}(y) + \tilde{b}_{\mu \nu}(x) Y_p^{(E)}(y)~.\end{aligned}$$ The AdS$_4$ coordinates are denoted by $x$ and greek indices, while their $S^7$ counterparts are $y$ coordinates and latin indices. The functions $Y(y)$, $Y_p^{(CE)}(y), Y_p^{(E)}(y)$ are spherical harmonics on $S^7$ that are respectively a scalar, a coexact 1-form, and an exact 1-form.

The $2$-tensor $\tilde{b}_{\mu \nu}(x)$ is the coefficient of $Y^{(E)}(y) = dY(y)$ which is effectively a scalar on $S^7$ so there is one of these modes, while $b_{\mu \nu}(x)$ is the coefficient of $Y^{(CE)}(y) = {}^*dY(y)$ which is effectively a transverse vector on $S^7$ with six modes. Thus there is a total of $1+6=7$ $2$-tensors in the effective 4D theory as one would also expect from toroidal compactification of 11D supergravity to 4D. Classically these seven antisymmetric tensors can be dualized to seven scalars but in view of quantum inequivalence this must be done with care.

The $3$-form tensor $b_{\mu\nu\sigma}(x)$ is the coefficient of the ordinary spherical harmonic so there is just one of these fields in four dimensions. A massless $3$-form has no propagating degrees of 