A.1 𝒩 = 2 type IIA/B superspace

In components, 𝒩 = 2 type IIA/B supergravities describe the dynamics of the gravity supermultiplet. The latter contains

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 M Z involving both bosonic m x and fermionic 𝜃 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

Φ(x, 𝜃) = ϕ(x) + 𝜃αϕα (x ) + ... (515 )
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 5View Image.

The superspace formulation of the 𝒩 = 2 type IIA/B supergravity multiplets is as follows:

  1. Given the presence of fermions, it is natural to work in local tangent frames. Thus, instead of using the metric variables gmn (x), one works in terms of bosonic vielbeins a E m(x). These are then extended to a supervielbein A EM (x,𝜃), where M = {m, α } stands for the superspace curved indices, whereas A = {a, α} describes both flat bosonic and fermionic tangent space indices. EAM(x, 𝜃) already includes the gravitino Ψm as a higher-dimension component in its fermionic 𝜃 expansion.
  2. One extends all remaining bosonic fields to superfields with the same tensor structure, i.e., B2 = 1Bmn (x)dxm ∧ dxn 2 is extended to B2 = 1BAC (x,𝜃 )EA ∧ EB 2, where A M A E = dZ EM, and similarly for all other fields, including the dilaton.

The following discussion follows closely Section 3 in [141Jump To The Next Citation Point]. 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:

A A A B A T = DE ≡ dE + E ∧ ωB , (516 ) RAB = d ωAB + ωAC ∧ ωC B. (517 )
The covariant derivative D is defined in terms of a Lorentzian connection one-form B ωA, but in type IIB, it includes an additional U(1) connection defined on the coset space SU (1,1)∕U (1) where the set of type IIB scalars live [309Jump To The Next Citation Point]. These superspace torsion and curvature forms satisfy the Bianchi identities
DT A = EB ∧ R A, (518 ) B DRAB = 0. (519 )

The first of the constraints I was alluding to before is the Lorentzian assumption. It amounts to the conditions

ω β-= 0 = ω b ⇒ R β-= 0 = R b. (520 ) a α- a α-
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. (514View Equation) 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 H3, the NS-NS super-three-form, by Rn, the RR super-n-forms, and define them as

10 H = dB , R = eB2∧ d(e−B2 ∧ C) ≡ ⊕ R , (521 ) 3 2 n=1 n
where I introduced the formal sum over all RR gauge potentials by ⊕9 C ≡ n=0 Cn and proceeded analogously for their field strengths51. These obey the Bianchi identities
dH3 = 0, (522 ) dR − R ∧ H = 0, (523 )
and are invariant under a set of gauge transformations leaving the supergravity Lagrangian invariant
δB2 = d λ1, (524 ) B2 δC = e ∧ d μ. (525 )
Since the Bianchi identity (523View Equation) allows one to set either the even or odd RR forms to zero, this reproduces the well-known statement that on-shell type IIA [136] and IIB [309] 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:
T c = 2iΓ c , T c = 0 , (526 ) αβ αβ aβ γ- 3- Γ γ- 1- a γ- IIA: Tαβ = 2δ(α-Λβ) + 2(Γ â™¯)(α (Γ â™¯Λ)β) − 2(Γ a)αβ(Γ Λ ) (527 ) 1 + (Γ aΓ â™¯)αβ(Γ aΓ â™¯Λ )γ +-(Γ ab)(α-γ(Γ abΛ)β), (528 ) γ γ 4 γ IIB: Tαβ--= − (J)(α-(JΛ )β) + (K )(α-(K Λ)β) (529 ) 1 1 + -(Γ aJ )αβ(Γ aJΛ )γ −-(Γ aK )αβ (Γ aK Λ)γ-, (530 ) 2 2 H αβγ = 0, (531 ) ϕ IIA: Ha βγ = − 2ie 2(Γ â™¯Γ a)βγ , (532 ) ϕ2 Habγ = e (Γ abΓ â™¯Λ)γ-, (533 ) IIB: H = − 2ieϕ2 (K Γ ) , (534 ) aβγ a βγ Habγ = e ϕ2(Γ abK Λ)γ , (535 ) - -- R (n)αβγA1...An−3 = 0 , (536 ) -- n−5 n IIA: R (n)a1...an−2αβ = 2ie 4 ϕ(Γ a1...an−2(Γ â™¯)2)αβ , (537 ) n − 5 n−5 n R (n)a1...an−1α-= − -----e 4 ϕ(Γ a1...an−1(− Γ â™¯)2Λ )α, (538 ) 2n−5 n−1 IIB: R(n)a1...an− 2αβ = 2i e-4-ϕ(Γ a1...an−2K-2-I)αβ , (539 ) - - R (n)a1...an−1α-= − n −-5en−45ϕ(Γ a1...an−1K n−21IΛ)α-. (540 ) 2 Λα = 1∂αϕ . (541 ) -- 2 --
Here, Γ â™¯ = Γ 0Γ 1...Γ 9 stands for the 10-dimensional analogue of the γ5 matrix in d = 4, i.e., the chirality matrix, whereas K and J are SO (2) matrices appearing in the real formulation of type IIB supergravity [141Jump To The Next Citation Point]. In the last line, ϕ stands for the superfield containing the bulk dilaton, whereas Λα- has the appropriate dilatino as its leading component.

Even though the dual potential B6 to the NS-NS 2-form B2 does not explicitly appear in the kappa invariant D-brane effective actions reviewed in Section 3, its field strength H7 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

1 1 1 IIA: H7 = dB6 − -C1 ∧ R6 + -C3 ∧ R4 − -C5 ∧ R2, (542 ) 2 2 2 IIB: H7 = dB6 + 1-C0 ∧ R7 − 1C2 ∧ R5 + 1C4 ∧ R3 − 1C6 ∧ R1. (543 ) 2 2 2 2
By construction, these obey the constraints
− ϕ2 IIA: Ha1...a5αβ = 2ie (Γ a1...a5)αβ, (544 ) H = − e− ϕ2(Γ Λ) , (545 ) a1...a6α- ϕ a1...a6 α- IIB: Ha1...a5αβ = 2ie− 2(Γ a1...a5K )αβ, (546 ) ϕ Ha1...a6α = − e− 2(Γ a1...a6K Λ)α, (547 )
and the Bianchi identities
1 IIA: dH7 + R2 ∧ R6 − --R4 ∧ R4 = 0, (548 ) 2 IIB: dH7 + R1 ∧ R7 − R3 ∧ R5 = 0. (549 )

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