A.2 𝒩 = 1 d = 11 supergravity conventions

There is a similar discussion for 𝒩 = 1 d = 11 supergravity [154] whose gravity supermultiplet involves metric gmn(x), a three gauge field potential A3(x), or its Hodge dual A6(x), and a gravitino Ψm (x ). When embedding this structure in 𝒩 = 1 d = 11 superspace [153, 117], one uses local coordinates ZM = (xm, 𝜃) where now 𝜃 stands for an 11-dimensional Majorana spinor having 32 real components. As before, the superfield encoding information about both the metric and gravitino is the supervielbein A M A E = dZ EM, the superfield extension of the bosonic vielbein a Em. The notation is as before, with the understanding that the current bosonic indices, both curved (m ) and tangent space (a), run from 0 to 10. Furthermore, A3 (x ) is extended into a superfield 3-form A3 (x,𝜃) with superspace components A (x, 𝜃) BCD.

As in type IIA and B, it is natural to introduce the field strengths of these superfield potentials

R4 = dA3 , 1 R7 = dA6 + -A3 ∧ R4 , (550 ) 2
are gauge invariant under the abelian gauge potential transformations
δA = dΛ , 3 2 δA6 = dΛ5 − 1Λ2 ∧ R4. (551 ) 2

These superfields satisfy the set of constraints

a α- β a b β a 1- b c a T = − iE ∧ E--Γαβ + E ∧ E - Tbβ + 2 E ∧ E T bc, (552 ) 1 1 R4 = -Eb-∧ Ea- ∧ Eα-∧ E β(Γ ab)αβ +--Ea-∧ Eb-∧ Ec-∧ EdRdcba, (553 ) 2 4! R = 1-Ea1-∧ ...∧ Ea5-∧ E α-∧ E β(Γ ) + 1-Ea1 ...Ea7R . (554 ) 7 5! a1...a5 αβ 7! a7...a1
These allow one to establish an equivalence between this superspace formulation and the on-shell supergravity component one. They are also crucial to proving the kappa symmetry invariance of both M2 and M5-brane actions.


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