- Type IIA: its bosonic sector contains metric , dilaton , NS-NS 2-form , RR potentials , whereas its fermionic counterparts includes the dilatino and the gravitino .
- Type IIB: its bosonic sector contains metric , dilaton , NS-NS 2-form , RR potentials , whereas its fermionic counterparts includes the dilatino and the gravitino .

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

whose components include the physical fields listed above. For an extensive and pedagogical introduction to the superfield and superspace formulation in supergravity, see [491].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:

- Given the presence of fermions, it is natural to work in local tangent frames. Thus, instead of using the metric variables , one works in terms of bosonic vielbeins . These are then extended to a supervielbein , where stands for the superspace curved indices, whereas describes both flat bosonic and fermionic tangent space indices. already includes the gravitino as a higher-dimension component in its fermionic expansion.
- One extends all remaining bosonic fields to superfields with the same tensor structure, i.e., is extended to , where , and similarly for all other fields, including the dilaton.

The following discussion follows closely Section 3 in [141]. 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:

The covariant derivative is defined in terms of a Lorentzian connection one-form , but in type IIB, it includes an additional connection defined on the coset space where the set of type IIB scalars live [309]. These superspace torsion and curvature forms satisfy the Bianchi identitiesThe first of the constraints I was alluding to before is the Lorentzian assumption. It amounts to the conditions

This guarantees the absence of non-trivial crossed terms between the bosonic and fermionic components of the connection and curvature in superspace. Conceptually, this is similar to the condition described in Eq. (514) in the superembedding formalism [460].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 as

where I introduced the formal sum over all RR gauge potentials by and proceeded analogously for their field strengthsEven 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 [141]. For completeness, I include its definition below

By construction, these obey the constraints and the Bianchi identities
Living Rev. Relativity 15, (2012), 3
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