Gravity and Strings

16.2 Romans’ massive N = 2A, d = 10 supergravity 465
field for the NSNS 2-form ˆB which transforms its kinetic term in the action into a conventional
mass term m2 ˆB 2 , and is the reason why this theory is called massive.
One may wonder how some of the fields of the supergravity multiplet can become massive
while the theory is formally invariant under N = 2 local supersymmetry since linearly
realized supersymmetry implies that all states in the same supermultiplet have the same
mass. The reason is not just gauge symmetry but also that supersymmetry is broken in this
theory. There are two ways to see this: on the one hand, a local supersymmetry transformation
can be used to gauge away one dilatino and give mass to one gravitino; on the other
hand, the most (super)symmetric vacuum of this theory, which is not Minkowski spacetime
but the D8-brane, breaks half of the supersymmetries12 as well as some of the isometries of
Minkowski spacetime. Romans’ massive N = 2A, d = 10 supergravity can be interpreted
as the effective-field theory of type-IIA superstrings with an open-string sector associated
with a D8-brane that breaks translation invariance and supersymmetry [782].
There are good reasons to interpret the mass parameter m as another (0-form) RR field
strength,
ˆG (0) ≡ m. (16.75)
A 0-form field strength has to be constant due to the Bianchi identity d ˆG (0) = 0. It can
be dualized (on-shell) into a 10-form field strength ˆG (10) whose equation of motion is this
Bianchi identity and whose Bianchi identity also follows the general rule Eqs. (16.57) and
implies the existence of a RR 9-form Ĉ (9) , which must be non-trivial in order to have
⋆ ˆG (10) =−ˆG (0) =−m.ARR9-form potential was required by string theory since the type-
IIA theory admits all even p Dp-branes, which couple to RR (p + 1)-form potentials and
in massless N = 2A, d = 10 supergravity there is no potential for the D8-brane. Romans’
theory describes the effective type-IIA string theory in the presence of D8-branes. The
trouble with the 9-form potential is that it does not have dynamical degrees of freedom
and, if we include it in the form of a mass parameter, there is a D8-brane and if we do
not include it, there is not a D8-brane, whereas, for lower-rank potentials, the same theory
admits solutions with and without branes. 13
The 11-dimensional origin of the D8-brane and its associated mass parameter are unknown,
although there are arguments based on the superalgebras of N = 2A, d = 10 and
d = 11 supergravity that support the idea that there is a nine-dimensional extended object
in d = 11 (the M9-brane discussed in [123, 142], also known as the KK9M-brane [666]),
which could also be associated with the (9 + 1)-dimensional boundaries of the Hoˇrava–
Witten construction discussed on page 16.4.
The supersymmetry transformation rules are given by Eqs. (16.70), where the sums have
to be extended up to n = 5toinclude ˆG (10) and the new field strengths have to be used. The
same is true for the expression for ˆH (7) , Eq. (16.70).
12 Actually, the mass of the KR 2-form should be determined in the D8 background. The Stückelberg mechanism
for higher-rank form potentials underlies many gauged higher-dimensional supergravities, but there
are cases in which the theory admits a maximally supersymmetric AdS vacuum with respect to which the
forms are massless in spite of the explicit “mass terms” they have in the action.
13 Actually, the theory can be generalized slightly, admitting the possibility that ˆG (0) is only piecewise constant,
which is equivalent to the introduction of sources for the dual Ĉ (9) potential which are D8-branes
placed at the discontinuities of ˆG (0) [118, 133], but we will not consider this generalization here.