Both theories differ in the chiralities of their fermionic sectors and the dimensionality of their RR gauge potentials. Furthermore, the field strength of the RR 4-form potential in type IIB is self-dual.
To make the local supersymmetry of this component formalism manifest, one proceeds as in global supersymmetry by introducing the notion of superspace and superfields. The theory is defined on a supermanifold with local coordinates involving both bosonic and fermionic ones. The latter have chirality properties depending on the theory they are attached to. The physical content of the theory is described by superfields, tensors in superspace, defined as a polynomial expansion in the fermionic coordinates.
A general feature of this formalism is that it achieves manifest invariance under supersymmetry at the expense of introducing an enormous amount of extra unphysical degrees of freedom, i.e., many of the different components of the superfields under consideration. If one wishes to establish an equivalence between these superspace formulations and the standard component ones, one must impose a set of constraints on the former, in order to consistently, without breaking the manifest supersymmetry, reproduce the on-shell equations of motion from the latter. This relation appears schematically in Figure 5.
The superspace formulation of the type IIA/B supergravity multiplets is as follows:
The following discussion follows closely Section 3 in . As in Riemannian geometry, we can describe the geometry of a curved background in terms of a torsion and curvature two forms, but now in superspace:. These superspace torsion and curvature forms satisfy the Bianchi identities
The first of the constraints I was alluding to before is the Lorentzian assumption. It amounts to the conditions.
Some of the additional constraints involve the components of the super-field strengths of the different super-gauge potentials making up the superspace formulation for type IIA/B introduced above. Denote by , the NS-NS super-three-form, by , the RR super--forms, and define them as51. These obey the Bianchi identities  and IIB  supergravities contain even and odd RR field strengths, respectively. To match the full on-shell supergravity formulation in standard components one must impose the following further set of constraints: . In the last line, stands for the superfield containing the bulk dilaton, whereas has the appropriate dilatino as its leading component.
Even though the dual potential to the NS-NS 2-form does not explicitly appear in the kappa invariant D-brane effective actions reviewed in Section 3, its field strength is relevant to understand the solution to the Bianchi identities in type IIB, as explained in detail in . For completeness, I include its definition below
Living Rev. Relativity 15, (2012), 3
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