On the geometry of forms on supermanifolds

Remarks on the Integral Form of D=11 Supergravity

“…which is a Lorentz singlet and explicitly depends upon 9 θ's. Then, the measure [Dλ] = d 23 λµ(λ, θ) is defined such that d 32 θd 23 λµ(λ, θ) Ω (7) (λ, θ) = 1 (29) where d 32 θ is the conventional Berezin integral and…”

“…Appendix: A Brief Review on Integral Forms In this appendix, we want to recall the main definitions and computation techniques for integral forms. For a more exhaustive review or for a more rigorous approach to integral forms we suggest [27,28,29]. We consider a supermanifold SM (n|m) with n bosonic and m fermionic dimensions.…”

Novel Free Differential Algebras for Supergravity

“…The presence of a naïve integral form in the spectrum tells us that we must build the integral form to construct a consistent action and its equations of motion [20][21][22][23][24]. One can relate this non-trivial integral form (or its potential) as a reflection of the existence of the Berezianian (see [30]), and the fact that there is a single non-trivial integral form seems to indicate that the Berezianian is a truly essential ingredient in the supergravity realm.…”

“…For a comprehensive look at various examples, please refer to the citation [25]. Additionally, the citation [30] can be consulted for a thorough mathematical derivation. The latter source explains the distinction between the differential d that operates on superforms and d K (known as the Koszul differential) that operates on the integral form.…”