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 [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 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 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
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 [141]. For completeness, I include
its definition below
http://www.livingreviews.org/lrr-2012-3 |
Living Rev. Relativity 15, (2012), 3
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