Kaluza-Klein supermultiplet of 11D supergravity I am studying Adel Belali's review paper "M(atrix) theory: a pedagogical introduction" for my undergraduate thesis. In the third lecture, part one, we dimensionally reduce 11D supergravity to get type IIA SUGRA and retain the  128 bosonic degrees of freedom and 128 fermionic degrees from the original theory. I have gone through this calculation.
However, in lecture 3 part two, we do Kaluza-Klein compactification of 11D supergravity and get 256 massive degrees of freedom in addition to the 256 massless degrees of freedom. Each massive KK state has a mass $\frac{n}{l_{s}g_{s}}$. My question is: how do these 256 massive degrees of freedom come about?
 A: Ten dimensional type IIA string theory is equivalent to M-theory with a compatified coordinate $x^{11}$. Two important aspects of this correspondence should be highlighted.

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*The 256 massless degrees of freedom of eleven dimensional supergravity descend to the ten dimensional type IIA supergravity massless multiplet after compactification.

*The circle compactification produce new massive states. Namely the type IIA dilaton $\phi$ descending from $g_{11,11}$ and the $C^{1}$ Ramond-Ramond field identified as the vector $g_{\mu,11}$.

The new massive states are KK modes have mass $n/g\mathcal{l}_{s}$ in natural units. Any of those states of mass $n/g\mathcal{l}_{s}$ has $n$ unitis of $C^{1}$ RR charge because they were generated by the usual KK mechanism via $g_{\mu,11}$. Those KK modes are nonperturbative states as seen from the ten dimensional persepective because its energy scales as $1/g_{s}$; they are the $D0$ branes of type IIA superstring theory.
To discover why they contribute with extra 256 massless degrees of freedom you have two options. The first one is to recognize that $D0$ branes carry exactly the same quantum numbers (44 gravitons,84 components of the 3-form, and 128 gravitinos) of the eleven dimensional M-theory multiplet because they can be thought as supergravitons traveling in the $x^{11}$ direction in the $R \rightarrow \infty$ limit of the ten dimensional perspective. The other way to see this is to consider IIA strings attached to the $D0$ branes, and count their transverse degrees of freedom.
The conclusion is that, for each $n$, the KK modes of M-Theory compactified on a circle contribute with extra 256 degrees of freedom. Recall also that any state with $n$ units of $C^{1}$ RR photon charge is a bound state of $n$ states of one unit of RR charge.
References:

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*Introduction to M-theory page 13.

*TASI Lectures on Matrix Theory chapter 3.

