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In Type IIA/IIB string theory we can have various fluxes, such as the 3-form H-flux, and the various Ramond-Ramond fluxes in even/odd dimensions.

In 11D SUGRA, however, the field content seems to only contain the graviton, gravitino, and 4-form flux F. Why can't we have fluxes of other dimensions in 11D SUGRA?

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The field content of various SUGRAs is defined by the irreducible representation of super-Poincare algebra. For the reference, I would suggest to look at Appendix B "String theory", Polchinski, Volume 2. Specifically, $d=11$ is discussed in Appendix B.5.

The question "Why super-Poincare?" is answered in Section I of "Supersymmetry and Supergravity", Wess and Zumino.

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