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In this review of SUSY gauge theories and the $AdS-CFT$ correspondence, on Page-29, firstly the Supergravity action for $D = 11$ is written. The authors then say that the action for the Type IIA Supergravity can be derived by dimensional reduction.

Afterwards, they say that "There exists no completely satisfactory action for Type IIB theory, since it involves an antisymmetric field $A^+_4$ with self-dual field strength." Then they write down an "action" which is supplemented by a self-duality condition.

$$*\tilde{F_5} = \tilde{F_5}$$

where $\tilde{F_5}$ is given by

$$\tilde{F_5} = F_5 - \dfrac{1}{2} A_2 \wedge H_3 + \dfrac{1}{2} B \wedge F_3$$

So my questions are:

  1. Why is the method of getting the action by dimensional reduction not valid in the Type IIB case?
  2. Why is it true that there exists no action (by action, I mean one with no extra conditions like the self-duality we imposed in this case) for a self-dual antisymmetric field?

EDIT : I found the answer to my second question here, which specifies the cases when we cannot construct an action.

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