N = 1 supercurrents of elevendimensional supergravity
Accepted: May
= 1 supercurrents of elevendimensional
Katrin Becker 0 1 2
Melanie Becker 0 1 2
Daniel Butter 0 1 2
William D. Linch III 0 1 2
Texas A 0 1 2
M University 0 1 2
0 Y that is chiral as a
1 College Station , TX 77843 , U.S.A
2 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy
Elevendimensional supergravity can be formulated in superspaces locally of the form X Y where X is 4D N = 1 conformal superspace and Y is an arbitrary 7manifold admitting a G2structure. The elevendimensional 3form and the stable 3form on Y de ne the lowest component of a gauge super eld on X super eld on X. This chiral eld is part of a tensor hierarchy giving rise to a superspace ChernSimons action and its real eld strength de nes a lifting of the Hitchin functional
Extended Supersymmetry; MTheory; Supergravity Models; Superspaces

HJEP05(218)
on Y to the G2 superspace X
Y . These terms are those of lowest order in a superspace
Noether expansion in seven N = 1 conformal gravitino super elds
. In this paper, we
compute the O( ) action to all orders in the remaining
elds. The elevendimensional
origin of the resulting nonlinear structures is parameterized by the choice of a complex
spinor on Y encoding the o shell 4D N = 1 subalgebra of the elevendimensional
super
Poincare algebra. Keywords: Extended Supersymmetry, MTheory, Supergravity Models, Superspaces
1 Introduction 2
Review and summary
Decomposition of the metric
Decomposition of the 3form ChernSimons and Kahler actions Additional gravitino super elds
4
5
6
7
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3
Extended supersymmetry and the gravitino supercurrent
The main result
The transformations and the gaugeforgauge symmetry of
Determining F and Gij from
transformations
Connecting with 11D superspace
The conformal supergravity supercurrent
Toward higherorder terms and a more covariant formulation
Outlook
A 4D superspace and supergeometry B
Useful variational expressions
B.1
The origin of covariantized transformations
B.2 Arbitrary variations of the ChernSimons and Kahler actions
realized in superspace [8] when the theory is truncated to four dimensions [9, 10]. As with
all theories representing more than nine supercharges, the superPoincare algebras close
only up to the component eld equations of motion [11]. That is, these supersymmetries
and the superspaces in which they are manifest are \onshell".
An alternative when only a part of the Poincare algebra is required to be manifest
is to represent only the supersymmetries associated to the subalgebra linearly. That is,
we may contemplate describing the elevendimensional supergravity theory as a bration
{ 1 {
by simpler superspaces. The simplest such choice (or at least the most welldeveloped by
far) is 4D N = 1 superspace. This superspace has many userfriendly features (e.g. o
shell, nitely many auxiliary elds, chiral representations) not the least of which is being
the most relevant phenomenologically. We therefore propose to study elevendimensional
supergravity in superspaces that are locally of the form X
Y where Y is any Riemannian
7manifold and X is a curved 4D N = 1 superspace. This program was initiated in [12{16].
In this paper, we extend the construction of elevendimensional supergravity in
superspace by deriving the supercurrents for the 4D N = 1 conformal supergravity and seven
conformal gravitino prepotentials (the spin2 and spin 32 parts) to all orders in the
remaining (spin
1) super elds. This is a necessary step in the determination of the action to all
orders in the gravitino super eld. Additionally, it elucidates the elevendimensional origin
of the conformal compensator elds: when supergravity is navely switched o , these are
super uous component elds required for the superspace embedding of the 3form
hierarchy. However in the superspace splitting X
Y , local superconformal symmetries emerge
in addition to those inherited from eleven dimensions. The consistency of our super eld
description hinges on the fact that precisely the seeminglysuper uous components
\compensate" for these fake symmetries. (They are their Stuckelberg elds.) We stress that the
conformal symmetry is in the X factor and arises as an artifact of the splitting.1
In section 2, we review the results of our previous work [12{16] on the embedding
of elevendimensional supergravity into o shell superspaces X
Y locally of the form
R4j4
R7. The component elds of elevendimensional supergravity consist of an elfbein,
a gravitino, and a 3form gauge eld. Decomposing these
elds in a (4 + 7)dimensional
split leads to the embedding of the 3form components into a collection of 4D N = 1
pform super elds with p = 0; : : : ; 3 that are qforms with q = 3
p in the additional
seven directions. The original elevendimensional abelian gauge symmetry with 2form
gauge parameter decomposes into a set of gauge transformations in 4+7 dimensions that
transform these super elds into each other. In addition, the pforms in the tensor hierarchy
are charged under the KaluzaKlein vector gauging di eomorphisms along Y . This gives
rise to a nonabelian gauging of the tensor hierarchy and a ChernSimonslike action which
we embedded in superspace [12, 13, 15].
One of the component elds in the tensor hierarchy is a [0; 3]form, that is a 0form in
spacetime X and a 3form in the internal space Y . It is naturally contained in the bottom
component of a chiral multiplet; since this bottom component is required by supersymmetry
to be complex, the pseudoscalar [0; 3]form is naturally paired with a scalar [0; 3]form
which is invariant under the abelian gauge symmetry. This object de nes a symmetric
bilinear form that can be taken to be positivede nite for generic 7manifolds Y [21]. As
such, it de nes a Riemannian volume on Y , and the super eld containing it de nes a
Kahler potential on the superspace. Modi ed by a certain function of another of the tensor
hierarchy super elds, this de nes a second N = 1 supersymmetric action which, together
with the ChernSimons action, describes the dynamics of elds with 4D spin
1Attempts have been made to de ne the superconformal symmetry directly in elevendimensional
superspace [17, 18], and to use it to represent all 32 supersymmetries o shell [19, 20].
{ 2 {
HJEP05(218)
A peculiar feature of this action is its invariance under global 4D N = 1
superconformal transformations, which include both scale (Weyl) and chiral U(
1
)R transformations.
It was argued in [14] that this is naturally enhanced to a local superconformal symmetry
when coupling to N = 1 supergravity. It was shown in [16] that the coupling to N = 1
supergravity and the additional seven gravitino multiplets could be achieved at the
linearized level  that is, considering the action quadratic in
uctuations about an onshell
4D Minkowski background with an arbitrary internal manifold of G2 holonomy and
vanishing 4form
ux. The explicit N = 1 superspace action was given and its component
action was shown to match that obtained directly from 11D.
Our goal in this paper is to proceed beyond the approximations of [14, 16] by
constructing the coupling of both N = 1 conformal supergravity and additional seven
gravitino super elds to the nonlinear action of the elds of lower 4D spin. Speci cally, we will
expand in super elds (denoted Ha and
i below) containing component spins
32 but
keeping the exact nonlinear dependence on the remaining super elds at each order. To
lowest order, this was done in reference [14] and checked to reproduce the nonlinear scalar
potential upon component projection. In section 3, we extend this result to the leading
order in the seven gravitino super elds
i by constructing the complete supercurrent J i
(cf. (3.8)), and motivate its derivation with a careful analysis of the symmetries of the
conformal gravitino super elds. The intricate compensator mechanism associated to this
superconformal symmetry has a strikingly simple interpretation when it is derived from
eleven dimensions. As we show in section 4, reducing the 11D N = 1 superspace frame to
4D N = 1 can be done in di erent ways parameterized by a complex scalar and a complex
spinor of SO(7). By studying the gauge transformations of this parameterization, we
discover that the bilinears of the spinors (dressed appropriately with the modulus and phase of
the scalar) reproduce the transformation rules de ning the superconformal compensators.
With the gravitino couplings thus understood, we turn in section 5 to their
superconformal graviton analog. Similarly to the superconformal gravitino multiplet, our analysis
has been perturbative in the Y dependence of the 4D N = 1 superconformal graviton:
in references [14, 15] the 4D N = 1 SG part was treated nonlinearly as a Y independent
background. In reference [16], Y dependent uctuations were studied but only to quadratic
order in all elds. In section 5, we begin to address this point by giving the
transformations of the gauged tensor hierarchy elds under the 4D N = 1 superconformal symmetry
to the rst nontrivial order, now with Y dependent gauge parameter. We then derive
the complete 4D N = 1 conformal supercurrent by requiring invariance under these
extended symmetries.
This completes the superspace construction to lowest nontrivial order in the spin2
and 32 components including dependence on all eleven dimensions but treating the spin
1
components exactly. To go beyond this order in spin
elds requires understanding
the nonlinear terms in the gravitino expansion. In section 6 we take the rst step in this
3
2
direction by constructing modi cations of the hierarchy
eld strengths that are invariant
under part of the seven extended superconformal symmetries.
{ 3 {
We begin by reviewing our previous results [12{16] on the description of elevendimensional
.
supergravity in terms of superspaces of the form X
Y where X is a 4D N = 1
supermanifold and Y is a real 7manifold. Locally X is of the form R4j4 with coordinates (xm;
;
and indices m = 0; : : : ; 3 and
;
= 1; 2 from the middle of the alphabets. Following
the early/late convention, tangent indices are taken from the beginnings (e.g. a, , and
). Local coordinates on Y will be denoted by yi with i = 1; : : : ; 7; we will generally not
need tangent indices for Y . The body of X (i.e. its bosonic part) will be denoted by X
and elevendimensional indices on X
Y will be denoted in bold so that, for example,
and 3form (section 2.2) and their superspace gauge transformations, eld strengths and
Bianchi identities. With these ingredients, we build an invariant action consisting of a
superspace volume term and a ChernSimonslike term (section 2.3). We conclude our review
by introducing the gravitino super elds and de ning the gravitino expansion (section 2.4).
2.1
Decomposition of the metric
Let's rst discuss the elds that arise from decomposing the 11D metric, as these will play a
role in de ning the covariant derivatives in 4D N = 1 superspace. We employ the standard
KaluzaKlein decomposition
(2.1)
(2.2)
gmn =
gmn + AmiAnj gij Amj gji :
!
gij Anj
gij
A priori, the 4D metric gmn, the KaluzaKlein gauge eld Ami, and the Y polarizations of
the metric gij each depend on all eleven coordinates.
The 4D metric gmn is encoded in a real super eld H _ = ( a) _ Ha with a linearized
gauge transformation
H _ = D _ L
D L _ :
This de nes it as an irreducible superspin 32 representation: at the component level it
contains the spin2 polarizations of the frame ema, the N = 1 gravitino
m , and an auxiliary
vector eld dm. The gauge symmetry (2.2) actually encodes local N = 1 superconformal
transformations, so that the frame is de ned up to an overall Weyl rescaling, the gravitino
is de ned up to a shift in its spin1/2 part corresponding to the action of the special
superconformal Ssupersymmetry, and dm is de ned up to chiral U(
1
)R transformations for
which dm is the gauge eld.
In [16], the super eld Ha appeared explicitly, but this quickly becomes unwieldy when
going beyond quadratic order. It is usually much simpler to employ a curved
WessZuminotype superspace where the prepotential Ha is encoded in the supervielbein EM A. The
natural superspace to employ is 4D N = 1 conformal superspace [22], where the N = 1
superconformal symmetry is explicitly gauged and which describes precisely (in
supergeometric language) the eld content of Ha. However, there is a caveat to employing this
{ 4 {
superspace: We have to assume that the supervielbein (and its prepotential Ha) do not
depend on y, since in the language of [22], the supervielbein is gauge invariant under any
internal gauge symmetries. In the context of our split spacetime, these include the GL(7)
di eomorphisms of Y . There is no technical obstruction to developing a superspace that
relaxes this condition, but it does not yet exist, and we will revisit this point in section 6.2
The nonabelian KaluzaKlein gauge elds Ami are described by a real unconstrained
prepotential Vi. Just as with the supergravity prepotential, it is more convenient to use
a covariant description where the KaluzaKlein multiplet is encoded in superconnections
AM
i which covariantize the N = 1 superspace derivatives r, so that the superconnections
transform as
Ai = r i with i a real super eld describing internal di eomorphisms. The
KaluzaKlein connections deform the algebra of the conformal superspace derivatives by
introducing the curvature terms
i is a chiral eld strength obeying the standard conditions r _ W
i = 0 and
i = r _ W
_ i. Here Lv denotes the Lie derivative along the vector eld v 2 X(Y ).
interior product (contraction) with the vector eld v. In contrast to the supervielbein, the
connection A may depend on y.
The remaining Y polarizations gij of the metric are 28 real scalars from the point of
view of X. We will address their embedding into super elds presently.
2.2
Decomposition of the 3form
The elevendimensional 3form splits up into a \tensor hierarchy" of pforms,
Cmnp ! Cmnp ; Cmn k ; Cm jk ; Cijk
which are embedded into a tower of pform super elds. (The remaining 28 scalars from gij
will also be embedded in these elds.) Being forms also in the seven directions, they are
charged under the nonabelian gauge eld. The abelian part of the gauge transformation is
parameterized by the super elds ij (chiral), Ui (real), and
(chiral) encoding the
components of an elevendimensional super2form. Their weights are summarized in table 2.
The nonabelian part g = diff(Y ) acts by the Lie derivative with respect to the real scalar
super eld i
.
2Actually, this is not as constraining a scenario as it might seem. Because the N = 1 supergeometry
describes the metric only up to Weyl rescalings, if the external metric gmn(x; y) factorizes as gmn !
e'(x;y)gmn(x), with ydependence sequestered in a conformal factor, then the supervielbein can be taken
to just describe the yindependent piece. In realistic scenarios, this would describe the actual background
con gurations of interest. Then an explicit ydependent Ha could be introduced to describe general
ydependent uctuations about that background.
{ 5 {
(2.3)
(2.4)
X
i
Vij
ijk
V
i
Cmnp




2forms
vectors
scalars
auxiliaries
Cmn i






Cm ij

gmi
G
Hi


Cijk, Fijk
dX

dij
fijk
d
i
of the gauged ChernSimons super eld hierarchy. The bosons G, Hi, and seven of the Fijk can all
be removed by a choice of WessZumino gauge (cf. section 2.4).
The super elds of the hierarchy (not including the KK vector just described) are as
follows (we mention only the embedding of bosons, summarized in table 1):
Scalars. There are 35 chiral elds
ijk containing the 35 pseudoscalars Cijk from the
3form and 28 metric scalars gij . (The remaining seven scalars will be shown to be
pure gauge.) These transform under nonabelian internal di eomorphisms and under
the abelian tensor hierarchy gauge transformations as
Vectors. There are 21 real, unconstrained vector super elds Vij containing vector elds
It has two eld strengths Eijkl (chiral) and Fijk (real)
and
F =
(Vij is the vector multiplet prepotential we introduce next.) The lowest component
of Fijk is related to a Riemannian metric gij by
1
144
pg gij :=
klmnpqrFiklFmnpFjqr :
As we explain in detail in section 4, this metric is a generalization of the G2 structure
metric and Fijk that of a G2 structure for the internal manifold Y . It is not in general
closed or coclosed.
Cm ij of the 3form. These transform as
Their chiral eld strengths W ij
V = L V +
1 2
W
should not be confused with those of the KK vectors.
= L
2i
1
2i
{ 6 {
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
2forms. There are seven 2form gauge elds Cmn i in the chiral spinor super elds
i
(2.10)
(2.11)
(2.13)
(2.14)
(2.15a)
(2.15b)
(2.15c)
(2.15d)
(2.15e)
(2.15f)
!h( ; v) :=
r v +
_ r _ v +
r
v +
r _ _ v :
(2.12)
It contains an excess of seven real scalars in the bottom component of Hi, but these
will be shown to be pure gauge.
3form. The 3form Cmnp is encoded in a real scalar super eld X with gauge
transformation
and a \reduced" chiral eld strength
X = L X +
1
2i r
r _
_
!h(W ; U )
G =
41 r2X +
W
The bottom component of G is a complex scalar, which can be interpreted as a the
chiral compensator super eld of (modi ed) old minimal supergravity [23, 24] (also
known as 3form supergravity [25])]. Its phase can be eliminated by a choice of U(
1
)R
gauge and its modulus can be eliminated (or absorbed into the metric) by a choice
of Weyl gauge.
Being given explicitly in terms of the prepotential super elds, the eld strengths are
the solutions to the Bianchi identities
The eld strength Hi is real
where we de ne the shorthand
H =
r _
_
+
W
:
!h(W; V )
1
2
1
2i r W
0 =
2i
E
E
1 2
4 r r F =
41 r2H =
r _ G = 0:
W
W
expressing the fact that these forms are closed in the extended de Rham complex [15].
These rules and de nitions are compatible with local superconformal symmetry on X.
In particular, we can consistently assign scaling dimensions ( ) and U(
1
)R weights (w) in
addition to engineering dimension (d) to all the gauge parameters, prepotentials, and eld
strengths. These are summarized in table 2.
{ 7 {
G(X) H(
) W (V ) F ( ) W (V) L
.
constructing F ^ F in superspace. This was the approach taken in reference [15] where it
was shown that the composite superforms are given by
F = E + E F
W
= EW
4i r2(F r F )
H = E + E H + !(W; F )
G = EG +
W
W
1
2
4i r2(F H) :
ir F W F + ir _ F W _ F
(2.16a)
(2.16b)
(2.16c)
(2.16d)
These complicated composite elds satisfy the same Bianchi identities (2.15) as the eld
strengths after which they are named. (This apparently highly nontrivial fact is simply
the fact that the wedge product of closed forms is closed [15].) Because of these closure
relations, the invariant action can be written
SCS =
12LCS = i
1 Z
2
Z
d
2
Here E and E are the full superspace and chiral superspace volume densities, respectively.3
The superconformal weights of these measures can be found in table 3.
3The volume densities are further discussed in appendix A. They are yindependent as they are built
invariance under internal di eomorphisms.
purely from the curvatures themselves. The most general possibility is given by [16]
SK =
3 Z
2
d x
The ingredients are as follows (cf. section 2.2):
g(F ) = det(gij (F )) is the determinant of the Riemannian metric on Y obtained from
Fijk using (2.7). This factor is needed for the integrand to be a scalar density under
internal di eomorphisms [14, 21, 26].
G is the superconformal primary chiral super eld strength carrying the 4form
eld
strength Fmnpq along the fourdimensional spacetime. It has weights (3; 2; 1) (cf.
supergravity. Its presence ensures the proper conformal weight of the integrand.
F is (for now) an arbitrary analytic function of the Weylinvariant combination
x := jHj2 = (GG) 2=3 gij HiHj
(2.19)
of the 3form
eld strengths Fmnp i. Its precise de nition will be given later.
We will refer to SK as the Kahler action.4 One of our goals of this paper is to determine
F from invariance under the extended (nonmanifest) supersymmetry (cf. section 3.1).
2.4
Additional gravitino super elds
In addition to the above supergravitycoupled action, we must introduce gravitino
superelds that capture the dynamics of the seven additional spin3/2 elds and which make the
full action invariant under extended supersymmetry. These should be contained within an
unconstrained spinor super eld
i subject to a large gauge transformation that eliminates
most of its component elds. The goal is to determine that gauge transformation, both
for the gravitini and for the matter elds, and determine what constraints it places on the
function F (x).
In [16], we constructed the secondorder action for uctuations about a at Minkowski
background times a G2holonomy manifold Y with closed and coclosed 3form 'ijk(y) and
its corresponding G2holonomy metric gij (y). Two prepotentials took background values:
h i = i' and hXi = 2. The uctuation elds and the gravitino super eld were subject to
4Strictly speaking, this is a misnomer since a Kahler potential depends on chiral multiplets and vector
multiplets gauging their isometries. The deformation considered here also contains the tensor multiplets
(chiral spinors)
the linearized gravitino gauge transformations
0
1
1 ijk =
1Vij =
i =
1Vi =
i =
i + gij D
1
1
2i '~ijklD2 l
2i 'ijk( k
j
k
)
1
2
i
( i +
i) :
(2.20a)
(2.20b)
(2.20c)
(2.20d)
(2.20e)
i is a chiral spinor and i is an unconstrained complex super eld. Here we have
assigned a gravitino weight 1 to , , and , and denoted the above transformations by 0
and 1 corresponding to how they change the gravitino weight of the corresponding eld.
One expects on general grounds that the full nonlinear action takes the form of a
power series expansion in the gravitino super eld
i. Schematically,
S11D = S0 + S1 + S2 + O( 3) ;
where S0 consists of the zerothorder Kahler (2.18) and ChernSimons actions (2.17). The
extended supersymmetry transformations should act on the super elds schematically as
= 0 + 1 + 2 + : : :
where 0 is nonvanishing only for the gravitino and corresponds to the nonlinear
generalization of (2.20a). Then solving the equations S11D = 0 orderbyorder, that is,
1S0 + 0S1 = 0 ;
2S0 + 1S1 + 0S2 = 0 ;
etc.
we can determine the higherorder modi cations of the action as well as higherorder
modi cations to the supersymmetry transformations. Note that these transformations are
in nitesimal (thus rstorder) in
and , and so n
will be O( n
) while for any other
eld n will be O( n 1).
In this paper, we will be concerned with the rst step of the Noether procedure, that is,
determining the complete rstorder modi cation S1 to the action and the full contributions
to 1 for the matter elds and 0 for the gravitino super eld. The complete expressions for
these quantities are given in eqs. (3.4), (3.6), and (3.7) of the next section.
3
Extended supersymmetry and the gravitino supercurrent
We begin this section by summarizing our main result: the nonlinear extensions to the
linearized transformation rules (2.20) and the correction to the action to
rst order in
gravitino weight. Subsequently, we will motivate these results, discuss their underlying
physics, and sketch some of the derivations.
where r
0
r . Then the variation of
constituents
0 and V, so that
As we are employing a covariant framework for the KaluzaKlein gauge prepotential Vi,
it appears implicitly in almost all of our formulae. For example, the spinor covariant
derivatives r
and r _ are given as
r
= eiLV r0 e iLV = r
0
LA ;
r _ = e iLV r0_ eiLV = r _
0
LA _ ;
0 does not possess the KK connection. This means that covariantly chiral
superelds such as
must be understood as
= e iLV 0 where
0 is chiral with respect to
has two parts arising from the variation of the independent
; on top of this, it is convenient to introduce
additional pieces for its covariantized transformation. Similarly, additional terms are naturally
included in the covariantized transformations of the other super elds. We nd
(3.1)
(3.2)
(3.3a)
(3.3b)
(3.3c)
(3.3d)
are both
(3.4a)
(3.4b)
(3.4c)
(3.4d)
(3.4e)
V := V +
:=
:=
+ iL V
;
1
for the prepotentials of the tensor hierarchy. We emphasize that
and
chiral with respect to r. (Additional comments can be found in appendix B.1.)
In terms of these covariantized transformations, the rstorder extended SUSY
transformations are
9F 0 HmF npqHq(GG) 2=3 (GG)1=3 :
(3.5)
F ;
1
4
G
i
2
i
2
1
2
G
G
G
F +
;
1
2
i
2
i
2
G
:
H +
G
H ;
In the transformation rule for , we have introduced the composite 4form
pg F (GG)1=3
=
3!
1 pg ijklmnp F mnp(F + jHj2F 0)
The antisymmetric symbol ijklmnp is a tensor density with entries
1, and upper indices
in the second line of (3.5) have been raised with the metric gij (F ).
As we will explain presently, the zerothorder transformation of the gravitino super eld
turns out to be
0
i =
i + G Gij r
j + i W ij
j
;
where
i is covariantly chiral and
i is an unconstrained complex super eld, as in the
linearized theory. In the full theory, we see that the second term is dressed with a factor
of G as well as a complex rank2 tensor Gij (see eq. (3.12) for its de nition) and a third
term involving the 2form
contribution to the action is
eld strength W ij has appeared. The rstorder gravitino
gauges the seven supersymmetries not manifest in N = 1 superspace, J i can be
interpreted as the Noether supercurrent for these supersymmetries. One can easily check
that the rst line counters the
transformation of the Kahler term whereas the second
line counters the
transformation of the ChernSimons term.
Further requiring
invariance generates no new contributions to the supercurrent, but
instead determines the function F (x) (recall eq. (2.19)) in terms of
as the solution to the quartic polynomial
S1 =
1 Z
2
where the complex super eld J i is given by
F^ := F
2xF 0
4 F
x ^4 + F^3
1 = 0 :
Together these imply, for example, that F 0(x) = 112 F^2 and F (x) = 13 F^ + 23 F^ 2, so that
both F and its rst derivative can be expressed in terms of F^. Both F (x) and F^(x) possess
in nite series expansions:
F (x) = 1 +
x
12
x
2
144
+
;
F^(x) = 1
x
12
x
2
48
+
+
Both functions are monotonic. Whereas F (x) slowly increases without bound, F^(x) slowly
tends to zero.
Finally, the function Gij in the gravitino transformation (3.6) is given by
Gij = (GG) 1=3 ^ 1gij +
F
2i F^Fijk gkl Hl(GG) 1=3 :
It has ( ; w; d) = ( 2; 0; 0) and is a Hermitian matrix: the rst term is real and
symmetric, and the second term is imaginary and antisymmetric. These results appear rather
complicated, but there proves to be a great deal of structure that constrains them. In the
remainder of this section, we will motivate these results and sketch their derivations.
transformations and the gaugeforgauge symmetry of
Let us begin by justifying the
transformations. We have chosen to x the gravitino's
conformal and U(
1
)R weights as in table 2, and to identify
i with no additional factors
as in (3.6). Taking into account the engineering and conformal dimensions of the various
elds, it is easy to see that we cannot assign a
transformation to Vij , X, or Vi,
provided we expect only eld strengths to appear on the righthandside of the covariantized
transformations.
Now keeping in mind the linearized transformation rule (2.20d) for
i, the only
nonlinear modi cation consistent with chirality involves a factor of G correcting the conformal
and U(
1
)R weights. Incidentally, this is also why it is necessary to choose
i to have a
lower GL(7) index. If the index were raised, we would need to lower it with a chiral metric,
but no chiral metric exists.
appears only as r _
i
and engineering dimension arguments.
Considerations of various weights and chirality similarly restrict the
transformation
of the remaining prepotential
ijk to at most be of the form (3.4a). The coe cient can
be determined by requiring
invariance of the terms in the ChernSimons action that are
purely chiral  that is, terms that cannot be lifted to full superspace by eliminating an
overall r
2 factor. Because the supercurrent term (3.7) is a full superspace integral, the
purely chiral terms must cancel on their own, and this is only possible if
transforms as
in (3.4a). (To verify this, it helps to use the expression (B.5) for the general variation of
the ChernSimons action.)
The
transformations now uniquely determine the rstorder gravitino action (3.7).
Because these transformations can be written purely in the language of di erential forms,
knowing nothing of the metric gij or the function F , the Kahler and ChernSimons actions
must be canceled separately by their respective gravitino terms. The
rst line of (3.8)
exactly cancels the transformations of ijk and
i in the Kahler term, whereas the second
line of (3.8) can be seen to cancel the ChernSimons term. The only terms in S1 we might
not determine in this way are those that are invariant  that is, terms where the gravitino
. But no such terms can be written down by virtue of conformal
Similar arguments can be made to motivate the
transformations. There is no
transformation one can postulate for
i that is consistent both with chirality and its various
weights. The other prepotentials are more subtle. Let us focus rst on Vi. The
transformation given in (3.4e) matches in the linearized approximation and is the only
nonlinear possibility up to multiplication by an overall function of the weightless combination
jHj2 (2.19). To eliminate this possibility requires some physical insight. At the linearized
level, the gravitino super eld transformation (2.20a) is subject to a gaugeforgauge
symmetry whereby
can be shifted by an antichiral super eld. (This is necessary so that the
counting of the degrees of freedom is correct: an
without such a shift symmetry would
introduce an additional gauge invariance absent in the linearized theory.) The
transformation (3.4e) respects this gaugeforgauge symmetry, but any such jHj2 modi cations
would break it. That is, if
i is an antichiral super eld
i, the transformation (3.4e)
leads to 1Vi =
12 (G i + h.c.), which is the form of a chiral gauge transformation of V,
i =
i + i, see section 3.1 of [16]. A compensating chiral gauge transformation with
i = 12 G i leaves V invariant. A similar argument xes the transformations of Vij and X.
To
x the
transformation for
ijk and to uncover the full expression for the
transformation of the gravitino is more subtle. A practical approach is to rst require
invariance around a eld con guration with Hi = 0. An obvious ansatz here is to generalize
the linearized results to (introducing factors of G for weight)
(3.13a)
(3.13b)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
1
2
ijk =
term does not actually violate the gaugeforgauge symmetry, as one can counter the
chiral shift in
by a shift of
i involving W ij . In order for this chiral shift symmetry to
hold at higher order in Hi, the i W ij j term that we have added cannot be dressed by any
function of jHj2. So the most general expectation for the gravitino is the expression (3.6)
for some complex rank2 tensor Gij agreeing with (GG) 1=3gij when Hi vanishes. (The
exact expression (3.12) cannot be so simply determined.) Let us record here the explicit
form for the gaugeforgauge symmetry:
i = i
;
i =
G i
;
i =
iW ij j
;
r _ i = 0 :
We now have su cient information to completely determine the
ijk transformation
to all orders in Hi. Let's assume that the full transformation of
is
2 r
i 2 Zijkl l + 3 [iW jk]
for some unknown covariant tensor Zijkl. This must reduce to (3.13b) in the limit where H
vanishes. Now under the gaugeforgauge symmetry (including the appropriate nonabelian
transformation), we nd
i = gij G(GG) 1=3
r
j +
;
ijk =
2 r
i 2 (GG)1=3F~ijkl l +
i = gij G(GG) 1=3
r
j + i W ij j + O(H) :
canceled if
Thus,
This transformation in the
W
W
part of the ChernSimons action (2.17) can only be
i acquires the term i W ij j . (Here the results of appendix B are useful.)
ijk =
2
i lr2Zijkl + 3i lW [liW jk] + 4i l@[l ijk] :
This cannot be made to vanish since one cannot choose a covariant Zijkl to cancel the other
two terms. What one can do instead is to arrange for
ijk to generate a trivial symmetry
of the action by taking
ijk / ijklmnp l
S0 ;
mnp
so that S0 = 0 automatically. This requires
(pg (GG)1=3F )
2i F[ijkHl] :
The rst term is the variation of the Kahler term with respect to
. The F ^ H term
^ W
combines with the W
terms in (3.17) to give the variation of ChernSimons
action with respect to . This leads to the full expression (3.4a), and agrees (as it must)
with the result when Hi vanishes.
We have not yet justi ed the forms of F and Gij . These will come from requiring invariance
under
transformations, but we can already make a few comments about Gij . Because it
can only be built from Fijk, G, and Hi (no derivatives may appear if the action should be
twoderivative),
Gij = (GG) 1=3 c0(x)gij + c1(x)FijkgklHl(GG) 1=3 + c2(x)HiHj (GG) 2=3
(3.20)
where the form factors ci are complex functions of jHj2 (2.19). (The factors of GG give
the correct Weyl weights.) Spacetime parity further constrains c0 and c2 to be real and c1
Deriving the form factors and the expression for F is a relatively straightforward if
laborious task. A number of ingredients aid in this. Collected in appendix B.2 are the
general variations of the zerothorder actions. The variation of the ChernSimons action is
naturally written as (B.5) in terms of the covariantized variations of the potentials (3.4).
The Kahler term is more naturally written as (B.6) in terms of variations of the
eld
strengths themselves. These are derived from the variations of the prepotentials (including
the KaluzaKlein prepotential) and (focusing only on the
terms) are given by
1F =
2 LG F +
4 r
1 2 (GG)1=3
i
8 r
U
+
2 (F ^ H) +
r (F ^ H)
i
4 r
4 r
i 2 iHi G
i
2 r _ ir _ Hi G
i
W
W
G ;
2i W F ) + iG
W _ r _ F
1G =
1H =
+
i
i
i
1
2
2 (LG
+
2 LG H +
1
(W
W _ ) + c.c.
An analogous expression can be given for W ij , but it does not appear in the Kahler action
so it is not as useful. In casting the result in this form, we have pushed derivatives onto
as much as possible. Note that the rst term in each expression (3.21) corresponds to a
uniform internal di eomorphism, so this variation just leads in the Kahler action to a total
internal derivative.
A tractable approach to checking
invariance (which we followed to completion) is to
work orderbyorder in W ij and W
immediately, while the linear ones lead to the form factors (3.12)
i. The terms quadratic in these eld strengths cancel
c0 = F^ 1
; c1 = 6iF^ 1F 0 =
2i F^ ; c2 = 0 ;
(3.22)
HJEP05(218)
and conditions (3.10). The explicit calculation is not particularly enlightening, so we do not
reproduce it here. Instead, in section 6 we will brie y sketch a more covariant approach that
should lead to a simpli cation of this check and also permit computation of the gravitino
terms to higher order.
Because of the complexity of the actions and transformation rules, it is easy to lose
sight of an important fact: under very mild assumptions about the basic structure of the
and
transformations, the complete rstorder transformations of the matter elds and
the gravitino have been determined, as well as the function F in the Kahler term. It is clear
that this should not be otherwise  we are striving to describe 11D supergravity and that
theory is unique  but it is heartening to observe that the consistent Noether coupling
of the additional seven gravitini does indeed determine the action and transformations
uniquely. In the next section, we will show how to make the connection more transparent.
Connecting with 11D superspace
4
The
and
transformations associated with the extended supersymmetry of the gravitino
super elds describe a rather large gauge group consisting of a chiral spinor
and an
unconstrained complex super eld
. As we have reviewed in [16], these transformations
permit a WessZumino gauge condition on
where only
ve components survive: the
extended gravitini
vector ym i, and a spinor
i
.
5
m i, a complex antiselfdual antisymmetric tensor tab i, a complex
At the end of the day, two remnants of the super eld
transformations survive: the extended supersymmetry transformation
i corresponding
to the bottom component of
ij and a bosonic transformation with parameter r
under which ym i transforms as a gauge eld. Here let us denote this parameter zi and
normalize it as the weightless quantity
i
4
zi :=
(GG)1=3 2 i :
r
j
This bosonic gauge transformation played a critical role in arriving at the correct
linearized action. There are 16 extraneous scalar elds encoded in the N = 1 super elds
not present in 11D supergravity. Two of these (the bottom components of G) are Weyl
and U(
1
)R compensators, which are natural from the point of view of N = 1 superspace.
The other 14 turned out to be pure gauge degrees of freedom that could be removed by the
bosonic gauge transformation involving zi. While this leads to a consistent description of
the underlying physics, it is slightly puzzling. After all, we could have imagined descending
from eleven dimensions directly and de ning our four dimensional elds. How would these
14 degrees of freedom appear in that dictionary? In this section, we will answer this
question by descending directly from 11D superspace to 4D N = 1 conformal superspace.
Elevendimensional superspace was introduced in [2, 3] to describe (onshell) 11D
supergravity. (Our conventions here di er slightly to admit a closer connection to
normalizations and conventions used in 4D N = 1 superspace.) We denote 11D vector and spinor
5Actually,
contains only the gammatraceless part of the extended gravitini. The spin1/2 components
are encoded in some of the fermions in the matter elds.
(4.1)
indices with hats. An 11D vector decomposes as V a^ = (V a; V a) and a 32component
Majorana spinor decomposes as
^ = (
I ; _ I ) where
and _ are twocomponent chiral and
antichiral spinor indices for SO(3; 1), and I denotes an SO(7) spinor index, which can be
raised or lowered with IJ . The 11D charge conjugation matrix is real and antisymmetric,
C ^ ^
=
0
IJ
^ =
0
_ _ IJ
^C
^
!
Spinor indices are raised and lowered as
Spin(10; 1) gamma matrices ( a^)^ ^ to be pseudoHermitian, obeying
and
^ =
^
C ^^. We take the
ya^ =
a^
. These
where the SO(7) gamma matrices ( a)IJ are imaginary and antisymmetric.
The superspace is described by a supervielbein EM^
M^ A^
B^ valued in SO(10; 1) so that
M^ ^
^ = 14
M
^ a^^b( a^^b)^ ^ and
M^ ^^b =
M^ a^
^ = 0. The
constraints on the torsion and curvature tensors imply that the geometry is completely
onshell with no auxiliary
elds present. For example, the nonvanishing tangentspace
components of the torsion tensor T A^ = DEA^ are given by
A^ and structure group connection
T^ ^c^ = 2( c^)^ ^ ;
Ta^^b ^ =
1
84
( c^d^)^ ^
Ta^ ^^ =
D ^Fa^^bc^d^ ;
1
36 Fa^^bc^d^( ^bc^d^) ^^
1
288
( a^^bc^d^e^)
F ^bc^d^e^ ;
in terms of a super eld Fa^^bc^d^ which turns out to be the supercovariant 4form eld strength.
As a consequence of the Bianchi identities, F is covariantly closed. The components of the
superRiemann tensor RA^B^ C^
D^ are also completely determined by the Bianchi identities,
but we won't need them here. One can make the 3form apparent in superspace as well
by introducing a super 3form CM^ N^ P^ with super 4form
eld strength super eld F = dC.
Aside from its top component, which must be the same super eld Fa^^bc^d^ appearing above,
the only other nonvanishing component of F is F^ ^c^d^ = 2 ( c^d^)^ ^. The full super 4form
is then given by
F =
2
1 Ec^ ^ Ed ^ E ^
^
1
4!
^
^ E ( c^d^)^ ^ +
Ea^
^ E^b ^ Ec^ ^ Ed^ Fd^c^^ba^ :
(4.5)
Closure is straightforward to check using (4.4).
Let us now descend to 4D N = 1. The key question is how to identify the N = 1
gravitino 1form E
possibility is E I =
in terms of the 11D 1form E ^ = (E I ; E _ I ). The most general
I E
+
where
I is some complex SO(7) spinor, which may
depend on all coordinates, and the ellipsis denotes the other seven additional gravitino
connections, which we will ignore. Actually, it is convenient to factor out a modulus
from
I so that it is normalized to I I = 1. Thus, we identify
E I
!
I 1=2 E
+
and
E _ I !
I 1=2 E _ +
:
(4.6)
(4.2)
(4.4)
In order for the N = 1 torsion tensor to be canonically normalized, we must introduce a
factor of
into Ea, that is
Ea^
!
Ea!
Ea
is a conformal compensator because it introduces a new Weyl symmetry
has weight 1, Ea has weight
1, and E
has weight
1=2. Given the
constraint on the norm of I , any variation can be written as
I =
i! I + yi ( i)IJ J
and
I = i! I
yi ( i)IJ J ;
(4.8)
in terms of a complex GL(7) vector yi and a real parameter !. We have written the SO(7)
gamma matrices as i = Eia a, so that they possess a GL(7) index. The 14 components
of yi should evidently correspond to the complex gauge symmetry associated with zi. The
real parameter ! describes a local U(
1
)R transformation, because it can be absorbed by a
phase rotation E
= i! E .
We can esh out these statements by constructing explicit expressions for the
superelds G and Hi. These are naturally encoded as the lowest components of their
corresponding 4form and 3form eld strengths in N = 1 superspace [15]
F4 = Ec ^ Ed ^ E
^ E ( cd)
F3i = Ec ^ E
_
^ E i( c) _ Hi +
G + c.c. +
;
These should arise from the decomposition of the 11D 4form eld strength (4.5) under the
identi cations (4.4), (4.6), and (4.7). The lowest dimension term of (4.5) exactly reproduces
those above provided we identify
G = 2 3 ;
G = 2 3 ;
Hi = 2 (
i ) 2
:
Because we are being somewhat schematic with the reduction, we should probably not
trust these results beyond lowest component. In particular, they may develop
dependent
modi cations. But this suggests that the bottom component of Hi is associated with the
complexity of : if I = I (up to a phase), Hij would vanish. This is consistent with the
interpretation of [14] that this eld can be gauged away in the component theory.
We still need to identify how the internal metric coming from 11D supergravity, which
we denote gij , is related to gij (F ). It is clear these cannot be exactly the same because
gij (F ) transforms under zi while gij must be independent of . Indeed, under the zi part
of the
transformations we nd
gij =
pg =
1
3
3 gij H^kzk + H^(igj)kz
k
2 pg H^izi + c.c.
6i F 0 H^(iFj)klgkmH^mzl + c.c. ;
Here and below we use the Weylinvariant combination H^i := (GG) 1=3Hi. Whatever
the internal metric gij is, it must be invariant under zi transformations. To nd it, the
(4.9)
(4.10)
(4.11)
(4.12)
In addition to det g, there is another scalar zinvariant, G1=3G1=3 ^ 1, which carries
Weylweight 2. The only such invariant scalar in the 11D theory is the conformal compensator,
F
so we identify
2 = G1=3G1=3 ^ 1
F
=)
F^ = (
2 2
)1=3 :
This identi es F^ in terms of 2
can nd that
. Now from the explicit equations (4.10) and (4.14), one
x := jH^ j2 = 4F^ 4
4F^ 1
:
additional relations are useful (recall eq. (2.19)):
x = jHj2 =
4 (F^ 1 +
x)H^izi + c.c. ;
1
3
F^ =
31 F^ H^izi + c.c.
(4.13)
Using these results together with (4.10), one can check that there is (up to normalization)
only one symmetric rank2 tensor that is zinvariant and Weylinvariant. This should be
identi ed with the internal 7D metric,
gij = F^gij +
det g = det g F^4 :
(4.14)
HJEP05(218)
This is nothing but the quartic polynomial (3.10), here derived as an algebraic equation
when x and F^ are both expressed in terms of 2
.
To con rm these identi cations, we should verify that one can consistently write down
a relation between the parameters zi and yi in (4.8). G and Hi vary under (4.8) as
G = 2 3
+ c.c. =
(4.17)
We want to compare this to the transformation under 1
. For G, this is straightforward
from (3.21), but for Hi we must correct for the gravitino super eld. In the WessZumino
ij ! 0, r
ij =
G Gij r
2 j j + : : : by (3.6). With
gauge (cf. appendix C of [16]) r
this taken into account, we nd
This can be identi ed with (4.17) if
G = G (GG) 1=3 ziHi and
Hi =
2(GG)2=3Gij zj + c.c.
yi = 2
F
^ 4 zi
4
1 gij H^j H^kzk +
2 F
i ^ 2gii0 gkk0 Fi0jk0 H^kzj
:
For additional con rmation, let us try to identify the bottom component of Fijk. As
with Hij, this contains scalar
elds not directly present in 11D: 28 of its 35 degrees of
freedom arise from the internal metric via the G2 relation (2.7), but 7 degrees of freedom
remained. It turns out that Fijk has a remarkably simple interpretation expressed by
i
(4.15)
(4.16)
(4.18)
(4.19)
(4.20)
This can be checked by verifying that the yivariation of (4.20) matches the zivariation
in (3.21a) using (4.19). Further evidence is provided by checking that the G2 relation (2.7)
holds upon inserting (4.20) and the expression for gij in terms of
and gij from (4.14).
The holomorphic structure of Fijk suggests that one should identify the bottom
component of the chiral super eld
ijk as
ijk = Cijk
ijk ;
1
2
where Cijk is the 3form descending from 11D supergravity. Matching transformation rules
con rms this, which means that the 3form Cijk de ned by the real part of ijkj actually
di ers from Cijk, just as gij de ned from Fijk di ers from gij . In both cases, adopting the
gauge where
is real (equivalently, where Hij vanishes) they become equal.
Let us end on one particularly interesting result that we have not completely
understood. The Hermitian metric Gij appearing in the gravitino transformation possesses
an inverse
(
G
1)ij = F^ 2gij
41 F^2H^ iH^ j
2
i F ijkH^l :
Remarkably, it is this inverse, rather than Gij itself, which has an elegant interpretation in
11D. We nd simply
(4.21)
(4.22)
(4.23)
(
G
1)ij = F^ 1 gij +
ij
:
5
The conformal supergravity supercurrent
Until this point, we have been treating 4D N = 1 conformal supergravity as strictly
yindependent. We have also implicitly assumed that the supervielbein was invariant under
the extended supersymmetry transformations to the order we are working. That is, we
have assumed 1EM
A = 0 (equivalently, 1H _ = 0). This is to be expected, since the
component vielbein always transforms with secondorder gravitino weight, that is, e
into the component gravitino
with SUSY parameter . This means we have not yet
actually determined that the rstorder gravitino coupling is consistent with supergravity
at the nonlinear level.
To remedy this, we will introduce the prepotential super eld H _ to describe
ydependent uctuations around the yindependent background vielbein EM A. The schema
for introducing prepotentials to deform a background (non at) geometry can be found
in refs. [27, 28]. (See [29] for the particular case of N = 1 conformal superspace.) The
prepotential H _ is now subject to the gauge transformations
H _ = r _ L
r L _ ;
(5.1a)
where the superspace derivative r is de ned in the yindependent background EM A. One
must assign L transformations to the other potentials. The right choices can be
deter0V =
2i r2(L r F ) ;
L W
2i r2(L H) ;
L _ W _ + iL {
W F
iL _ {W _ F ;
0X = r (L G) + r _ (L _ G) + iL {
W H
iL _ {W _ H ;
0V =
L W
L _ W _ :
In addition, as in the linearized case [16, 30], we must assign an L transformation to
the gravitino,
1
mined following for example [29], and correspond to the (covariantized) transformations
i
2
1
2
We have labeled these by gravitino weight. In principle, each of the equations (5.1) may
possess higherorder gravitino modi cations on the righthand sides. We should also
mention here that L
itself possesses a certain gaugeforgauge symmetry where it can be
shifted by a chiral spinor super eld; this shift is countered in (5.1g), for example, by a
shift in
.
The rst order coupling of H _ to the nonlinear action is
SHa =
d x
where J _ can be interpreted as the supercurrent. We derive it directly by requiring gauge
invariance to lowest order in the L transformations. This is a long calculation that can be
split into two parts. The rst arises from the minimal coupling of H _ to the ChernSimons
action and is naturally written as a 6form,
(5.1b)
(5.1c)
(5.1d)
(5.1e)
(5.1f)
(5.1g)
(5.2)
(5.3)
(5.4)
J CS_ =
i
4 r F ^ r _ F ^ H
W
^ W _ ^ F
W
^ {W _
F ^ F
W _ ^ {W F ^ F
3 {W F ^ {W _
F ^ F ;
1
2
i
2
2
+
8
i
4 3! 3!
ijklmnp
r Fijkr _ FlmnHp :
or equivalently as a density
J CS_ =
1 pg W ij W _ klF~ijkl + i pg W ij W _ kFkij +
i pg W _ ij W
kFkij +2pg W
i
W _ j gij
1
+
The contribution from the Kahler term is more complicated and given by
J K_ = 2 [r ; r _ ] (GG)1=3pg(F^
3
2 F )
3
2
(GG) 1=3pg F 0Hi[r ; r _ ]Hi
(pg F (GG)1=3) [r ; r _ ]Fijk
+ (GG)1=3pg F^ r _ log(G=G)
3r _ (GG) 1=3pg F 0Hir Hi + c.c.
F 0(GG) 1=3 3HiW _ j W ij
3HiW
j W _ ij
6iHiW
j
W _ kFijk :
(5.5)
The two quantities J _ = J CS_ + J K_ , given above, and J i in (3.8) describe the two
supercurrents of 11D supergravity written in N = 1 language. J _ is the N = 1 conformal
supergravity supercurrent, and de nes the rstorder coupling to the ydependent
uctuation super eld H _ . Similarly, J i is the extended gravitino supercurrent, describing the
rstorder coupling to
i. When the covariant N = 1 super elds obey their equations of
motion, these currents are subject to the conservation conditions
r2J i = 0 ;
Gr (Gij J j ) =
iW _ ij J _ j ;
(5.6a)
(5.6b)
(5.6c)
These conservation equations are a direct consequence of the gauge transformations (3.6),
(5.1a), and (5.1g).
6
Toward higherorder terms and a more covariant formulation
The ability to couple
i and H _ to the nonlinear action at rst order provides a strong
check of consistency. Their couplings correspond to the N = 1 supercurrent and extended
supersymmetry supercurrents of 11D supergravity. The associated lowestorder extended
SUSY transformations completely determine the function F in the Kahler action and lead
to a number of consistency conditions. In this section, we will describe how these couplings
could be taken to all orders.
One key feature that was useful in determining the gravitino supercurrent J i was
the simplicity of the
part of transformations of both the gravitino (3.6) and the other
prepotentials (3.4). These are di erential form transformations involving neither the metric
gij nor the function F appearing in the Kahler action, and so the Kahler and ChernSimons
actions must be canceled separately. If this feature holds to all orders, it would mean that
in order for the Kahler action to be canceled by gravitino terms, it must be possible to
construct new
eld strengths Fijk, Hi, and G by introducing
modi cations of the old
eld strengths so that
invariance is manifest.
To lowest order, this is precisely how the Kahler part of the supercurrent (3.8) arises.
Indeed, the supercurrent (3.8) can be rewritten as
3i
2
144
J i =
+ W
i
i
2
1 ijklmnpFjkl (G r Fmnp
3 HmW np) :
!
3pg GG 1=3
F
(6.1)
Thus, one simply makes the following shifts in SK ,
G ! G = G + G{W
;
H ! H = H
F ! F = F +
^ W
i
2
2 r (G
) +
2 r _ (G
_ ) ;
_ ^ W _ ;
which are invariant to lowest order, and expands to rst order in
.
This observation cannot be extended simply by exponentiation. Remarkably, however,
it is possible to construct higherorder modi cations that ensure
invariance. For example,
(6.2a)
(6.2b)
(6.2c)
(6.3a)
(6.3b)
(6.3c)
(6.4a)
(6.4b)
(6.4c)
G
H
F
G = G{W
H =
F = +
+
i
2
4
2 r (G
^ W
+ O( 3) ;
G
16 r
2
4
i
4
i
1
G {W _ (r _
) + c.c. + O( 3) ;
^ H
8 r
^ r _ H
^ W
2V =
2
1 2
8 r (
^ H) :
_ G ;
correspond to modi ed invariant eld strengths, provided we introduce higherorder
transformations of the form6
Proceeding in this way, one determines the higherorder
transformations order by order.
To apply this logic to the ChernSimons action, recall that in the component 11D
theory one can write the integral R C ^ F ^ F over 11D spacetime M as the integral
R F ^ F ^ F over some auxiliary 12manifold whose boundary is M [31]. This can be
extended to N = 1 superspace by taking Y to be the boundary of an 8manifold Z and
integrating a super [4; 8]form on X
Z. The requisite 12form was computed in eq. (5.49d)
of ref. [15]. Being the superspace version of F 3, it involves only eld strength super elds
so that we can proceed to apply the procedure above.
Carrying out this program generates a large class
corrections necessary for
invariance. However, at the linearized level [16], there are quadratic terms involving r _
that are already
invariant, and we should expect corrections to these terms in the
nonlinear theory. In principle, they can be determined by requiring L and/or
invariance.
6There is some ambiguity in these transformations corresponding to the ability to make O(
2
) eld
rede nitions of
.
In this paper we have given the construction of elevendimensional supergravity in 4D
N = 1 curved superspace to
rst nontrivial order in the
elds with 4D spin
to all orders in the remaining elds. More precisely, we formulated a gravitino super eld
expansion of elevendimensional supergravity and solved it to leading and nexttoleading
order for the action and gauge transformations. The consistency of the construction relies
on a powerful set of local superconformal symmetries arising from the foliation of the
spacetime by N = 1 superspaces. This formulation is wellsuited to backgrounds in which
the spin
32 components are Y independent but otherwise arbitrary (assuming vanishing
vacuum values for the gravitini) such as warped compacti cations with
uxes.
32 and
The new local symmetry can be understood by comparing it to elevendimensional
superspace. Reduction of the 32 supersymmetries to 4 is parameterized by a complex
spinor. Dirac bilinears in this spinor de ne the expected G2 structure on Y , but additionally
we nd deformed chiral and linear super elds corresponding to conformal, U(
1
), and special
conformal compensators. In the super eld description, these compensators are mixed up
in multiplets containing physical elds. Because of this, interactions of the latter can be
determined exactly by requiring invariance under the gauge transformations of the former.
On the other hand, the gravitino super elds share compensating Stuckelberglike
super eld transformations with some of the physical elds for which the action is determined
exactly. Covariantizing under this part then introduces gravitino corrections to the eld
strengths appearing in there, which we constructed explicitly at nexttonexttoleading
order. What is more, the fourdimensional part of the superconformal symmetry appears
in the gravitino transformations through its Y dependence, again in a Stuckelberglike shift
that can only be canceled by similar logic. This time it involves the 4D N = 1 conformal
supergravity prepotential Ha.
As a rst step in this direction, we have explicitly constructed the 4D N = 1
supercurrent responsible for this mechanism at lowest order. However, working orderbyorder in a
this prepotential is vastly more complicated, since it couples to everything. Instead, one
might attempt a more covariant approach in which this eld does not appear explicitly.
This would be in close analogy to how we have treated the KaluzaKlein gauge
covariantly, hiding it in the superspace connection and manipulating only its eld strength
W explicitly. In such an approach, the new covariant object Xia replacing @iHa and
analogous to the aforementioned eld strengths would appear as the curvature components in
the superspace commutator [rA; ri] where rA is the conformal superspace connection [22].
This supergeometric approach is currently under investigation.
eld V
Acknowledgments
We thank Sunny Guha and Daniel Robbins for discussions and collaboration during the
early stages of this project. This work is partially supported by NSF under grants
PHY1521099 and PHY1620742 and the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University. We also thank the Simons Center for Geometry and
HJEP05(218)
Physics and the organizers of the September 2017 Workshop on Special Holonomy, where
results from this work were reported.
A
4D superspace and supergeometry
Our conventions for 4D N = 1 superspace follow [22], where N = 1 conformal superspace
was introduced for describing conformal supergravity. The 4D supermanifold is described
by local coordinates zM = (xm; ; _ ) and is equipped with a set of 1form connections
that gauge the N = 1 superconformal algebra. These are the supervielbein EM A, a spin
connection
connections FM
tive rA is given by
M ab, a U(
1
)R connection AM , a dilatation connection BM , Ssupersymmetry
and FM _ , and special conformal connections FM a. The covariant
derivarA = EAM
M abMba
BM D
AM A
FM
AKA
(A.1)
where Mab is the Lorentz generator, D is the dilatation generator, A is the U(
1
)R
generator, and KA = (Ka; S ; S _ ) collectively denotes the three special (super)conformal
connections. The algebra of these generators with each other and rA can be found in [22]
and matches the global N = 1 superconformal algebra with rA identi ed as the
supertranslation generator PA = (Pa; Q ; Q _ ). The presence of a nonvanishing superWeyl
tensor W
deforms the algebra by introducing curvatures in the (graded) commutators
[rA; rBg. While the lowest anticommutators are unchanged
fr ; r _ g =
2i( a) _ ra ;
fr ; r g = 0 ;
fr _ ; r _ g = 0 ;
a dimension3=2 curvature operator is introduced a la superYangMills,
[r ; r _ ] = 2
W _ ;
[r _ ; r _ ] = 2 _ _ W ;
where
W
i W
( ab)
Mba +
i
2 r W
S +
i
2 r _ W
( b) _ Kb :
The vectorvector curvature [ra; rb] can be found in [22]. The super eld W
mal primary (annihilated by KA) and chiral (annihilated by r _ ) and contains the curvature
is a
confortensors of conformal supergravity.
As discussed in [22] (see also [27] and [28] for the conventional formulations in N = 1
superspace), an invariant full superspace integral is built out of a scalar function L via
where E = sdet(EM A) is the superdeterminant (or Berezinian) of the supervielbein.
Superdi eomorphism invariance in superspace guarantees supersymmetry in components.
Chiral superspace integrals are built out of chiral super elds Lc via
Z
Z
d4x d4 E L ;
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
0 =
Z
B
B.1
for some V A, it follows that
Z
Z
Z
d4x d4 E
rAV A
FABKBV A ( )a ; (A.8)
d4x d4 E rAV A ( )a =
d4x d4 E FABKBV A( )a :
So if V A is not a conformal primary, there is a residual connection term left over. This
actually re ects the fact that in these cases, rAV A is not itself a gauge invariant Lagrangian.
Useful variational expressions
The origin of covariantized transformations
The tensor hierarchy that descends from the 3form of 11D supergravity is gauged by the
nonabelian KaluzaKlein connection. This is most easily described in di erential form
notation where the 11D exterior derivative decomposes as d11D ! D + @ +
F
.
D is
the covariant derivative in four dimensions, D = d
LA, where A is the KaluzaKlein
connection, @ is the internal derivative in seven dimensions, and F is the interior product
on an internal form index with the KaluzaKlein eld strength. The 4form
eld strength
G = dC descends to the set of ve eld strengths G[0;4];
; G[4;0] as
Arbitrary variations of these eld strengths involve varying both C[p;3 p] and the
KaluzaKlein vector, leading to
(A.7)
(A.9)
(B.1a)
(B.1b)
(B.1c)
(B.1d)
(B.1e)
(B.2a)
(B.2b)
(B.2c)
(B.2d)
(B.2e)
where E is the chiral superspace measure, see [22] for its de nition in superspace. In
both cases, the functions L and Lc must be conformal primaries and they must possess
appropriate Weyl and U(
1
)R weights; that is, L must have Weyl weight 2 and be U(
1
)R
neutral, while Lc must have Weyl weight 3 and U(
1
)R weight 2.
Full superspace integrals are related to chiral superspace integrals via
Z
d4x d4 E L =
4
E r2L :
Because of the presence of the special (super)conformal connections FM A, the standard
rule for a total covariant derivative is slightly modi ed. Using
G[1;3] = DC[0;3] + @C[1;2] ;
G[2;2] = DC[1;2] + @C[2;1] + F C[0;3] ;
G[3;1] = DC[2;1] + @C[3;0] + F C[1;2] ;
G[4;0] = DC[3;0] + F C[2;1] :
G[1;3] = D C[0;3] + @ C[1;2]
AG[0;4] ;
G[2;2] = D
G[3;1] = D
G[4;0] = D
C[1;2] + @ C[2;1] + F C[0;3]
C[2;1] + @ C[3;0] + F C[1;2]
C[3;0] + F C[3;1]
where
C[p;3 p] := C[p;3 p] +
AC[p 1;4 p]
(B.3)
are the covariantized transformations of the pforms. The relations (B.2) can be understood
as the general variation of the G[p;4 p] eld strengths consistent with the Bianchi identities.
A corresponding set of relations exist for the pform hierarchy written in N = 1
superspace. There the situation is more subtle because the connections A and C[p;3 p]
must be built from prepotential super elds. The superspace analogue of the set of
eld
strength variations (B.2) is
H =
G =
r ( V W
1
2 V r (W
1
V + i V
F
r _
i V W F )
2i W F ) +
r _ ( V W _ + i V W _ F )
1
2 V r _ (W _ + 2i W _ F ) ;
i V
H) +
W
i
L V
G :
V r F
L V W ; (B.4b)
(B.4c)
(B.4d)
E LCS;F + c.c. ;
(B.5)
2 SCS =
Z
d x
i
2
F ^
W
Y
4 E LCS;D +
G +
W
^ W
1
2
^ W
Z
d x
^ F ^
V r F +
W _ ^ F ^
V r _ F :
i
2
1
2
1
2
space piece,
[2ex] LCS;F =
where the covariantized transformations of the prepotentials was given in (3.4), analogous
B.2
Arbitrary variations of the ChernSimons and Kahler actions
The variation of the ChernSimons action with respect to the tensor hierarchy and
KaluzaKlein prepotentials can be decomposed in terms of a full superspace and chiral
super[2ex] LCS;D =
^ H + F ^ r W
+ 2r F ^ (W
i W F ) + c.c.)
^ F + c.c.) +
F ^ F +
H ^
V W F ^ r F +
F ^
V W _
F ^ r _ F +
F ^ rW F ^ F
1
2
1
For the Kahler term, it is easier to give its variation in terms of those of the covariant
objects directly,
2 SK =
LK =
Z
1
144
Z
d y
LK ;
1=3 pg F 0 gij Hi Hj ;
GG 1=3 pg F
2jHj2F 0
log GG
(B.6)
and then employ (B.4).
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
HJEP05(218)
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