Maximal supergravity in D=10: forms, Borcherds algebras and superspace cohomology

TL;DR

This paper derives the form structure of D=10 supergravity using superspace cohomology, showing consistency conditions for higher-rank forms and their relation to Borcherds algebras, with implications for string corrections.

Contribution

It provides a simple derivation of supergravity forms from supersymmetry and cohomology, linking form degrees to Borcherds algebra representations and exploring higher-rank forms beyond spacetime limits.

Findings

Higher-rank forms satisfy consistent Bianchi identities if physical fields do

Form degrees extend beyond spacetime, matching Borcherds algebra predictions

Thirteen-forms in IIB have non-trivial Bianchi identities despite being zero in supergravity

Abstract

We give a very simple derivation of the forms of $N=2,D=10$ supergravity from supersymmetry and $SL(2,\bbR)$ (for IIB). Using superspace cohomology we show that, if the Bianchi identities for the physical fields are satisfied, the (consistent) Bianchi identities for all of the higher-rank forms must be identically satisfied, and that there are no possible gauge-trivial Bianchi identities ($dF=0$) except for exact eleven-forms. We also show that the degrees of the forms can be extended beyond the spacetime limit, and that the representations they fall into agree with those predicted from Borcherds algebras. In IIA there are even-rank RR forms, including a non-zero twelve-form, while in IIB there are non-trivial Bianchi identities for thirteen-forms even though these forms are identically zero in supergravity. It is speculated that these higher-rank forms could be non-zero when higher-order string corrections are included.