Gravitational Duality Transformations on (A)dS4
arXiv:0704.0531v3  [hep-th]  14 Aug 2007
Preprint typeset in JHEP style - HYPER VERSION hep-th/yymmnnn
ILL-(TH)-07-02
Gravitational Duality Transformations on
(A)dS4
Robert G. Leigh
Department of Physics
University of Illinois at Urbana-Champaign
1110 West Green Street, Urbana, IL 61801-3080, USA
Email: rgleigh@uiuc.edu
Anastasios C. Petkou
Department of Physics
University of Crete
Heraklion 71003, Greece
Email: petkou@physics.uoc.gr
Abstract: We discuss the implementation of electric-magnetic duality transformations in four-
dimensional gravity linearized around Minkowski or (A)dS4 backgrounds. In the presence of a
cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynam-
ics is unchanged. We pay particular attention to the boundary terms generated by the duality
transformations and discuss their implications for holography.
http://arxiv.org/abs/0704.0531v3
mailto:rgleigh@uiuc.edu
mailto:petkou@physics.uoc.gr
Contents
1. Introduction and Summary 1
2. Action and Hamiltonian 2
2.1 The 3 + 1 split 4
2.2 Shifted Variables 7
2.3 Linearization 7
3. Linearized Gravitational Duality and Holography 9
3.1 Duality and Holography 9
3.2 Linearized gravitational duality 10
3.3 Linearized Constraints and Bianchi Identities 11
3.4 Connection with other known dualities 12
4. The Effect on the Boundary Theory 13
5. Conclusions and Outlook 13
6. Appendix: other duality mappings 14
1. Introduction and Summary
Duality has played an important role in our understanding of Yang-Mill theories and it is believed
that it will play an important role also in gravity and in higher-spin gauge theories. Indeed,
although it is less clear what could be the implications of duality for theories whose quantum
versions are still unknown, gravity and higher-spin gauge theories1 are intimately connected to
a quantum string theory where certainly duality plays a crucial role.
The recent advent of holography raises some intriguing questions for duality. For example
one may wonder what is the holographic image of a duality invariant spectrum, a duality trans-
formation or a possible quantization condition that usually duality implies for charges. Some of
these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic
duality of a U(1) gauge theory on AdS4 is responsible for a “natural” SL(2,Z) action on current
two-point functions in three-dimensional CFTs.2 Shortly afterwards it was shown in [3] that such
an SL(2,Z) action is intimately related to certain “double-trace” deformations in the boundary,
1For reviews of higher-spin theories see e.g. [1].
2See [4] and [5] for more recent works.
– 1 –
assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions
are strengthened by the fact that there exist models (e.g., see [6] and references therein) which
exhibit the required behavior. In particular, it was shown in [3] that certain ”double-trace”
deformations induce an SL(2,Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2)
currents. This has led to the Duality Conjecture of [3]: linearized higher-spin theories on AdS4
spaces possess a generalization of electric-magnetic duality whose holographic image is the natural
SL(2,Z) action on boundary two-point functions.
Surprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not
widely known by the time of this conjecture.3 Second order linearized gravitational duality was
discussed among other in [8, 9, 10, 11, 12]. More recently, the duality properties of linearized
gravity around flat space were studied in [13] and were further discussed in [14]. The duality of
linearized gravity around dS4 was later studied in [15].
In this note we present our calculations regarding the duality properties of gravity in the
presence of a cosmological constant. Having in mind applications to higher-spin gauge theories
we use forms and work in the first order formalism where duality is also manifested at the level
of the action [16]. Moreover, the first order formalism is relevant for applications of duality to
holography, since the correlation functions of the boundary theory are essentially determined by
the bulk canonical momenta (see e.g. [17]).
Our aim in this work is to formulate linearized first order gravity using suitable ”electric”
and ”magnetic” variables, in close analogy with electromagnetism. We find that this is pos-
sible only when the background geometry is Minkowski or (A)dS4. Then we implement the
standard electric-magnetic duality rotations. We find that, up to ”boundary” terms, the lin-
earized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter
the second order bulk equations of motion. Moreover, the duality rotation interchanges the
(linearized) constraints with the (linearized) Bianchi identitites. The ”boundary” terms have
important holographic consequences since they correspond to marginal ”double-trace” deforma-
tions [3] that induce the boundary SL(2,Z) action. In the Appendix we exhibit a modified
duality rotation that leaves the bulk Hamiltonian invariant and induces ”boundary” terms that
correspond to relevant deformations as in [3].
2. Action and Hamiltonian
Having in mind the extension of our results to higher-spin gauge theorieswe start from the
MacDowell-Mansouri form [18] of the gravitational action4
IMM =
ǫabcd
Rab ∧ Rcd + 2Λea ∧ eb ∧ Rcd + Λ2ea ∧ eb ∧ ec ∧ ed
, (2.1)
3An interesting formulation of first order duality for linearized gravity around flat space was presented in [7].
4We note I = −16πGNS, where S is the usually normalized gravitational action.
– 2 –
where a, b, ... are Lorentz indices. In this formalism, the vierbein ea and the spin connection ωab
are initially thought of as independent variables. The curvature 2-form is
Rab = dω
b + ω
c ∧ ω
Rabcde
c ∧ ed.
Varying the action with respect to ea and ωab, we find
Rab + Λea ∧ eb = 0 , (2.2)
T a = dea + ωab ∧ e
b = 0 . (2.3)
The relation to gravity is established via the vanishing torsion equation (2.3), which relates e
and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric
variables
Rµν −
Rgµν = +3Λgµν . (2.4)
and the scalar curvature is R = −d(d − 1)Λ = −12Λ. Note that our Λ is related to the
cosmological constant in its usual definition via Λcosm = −6Λ. Λ > 0 corresponds to AdS.
Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection.
Note that Λ has units (Length)−2. In the SO(3, 2)-invariant formalism, IMM arises from
IMM =
ǫABCDEV
ERAB ∧RCD , (2.5)
where V E is a non-dynamical 0-form field (that we take to have value V −1 = 1 to gauge back
to the SO(3, 1) formalism) and RAB is the curvature of Ω
B ≡ {e
a, ωab}. There are also quasi-
topological terms of the form
Itop =
RAB ∧R
RAB ∧ RACV
BV C (2.6)
that we could add to the action. In the stated gauge, this reduces to
Itop =
P2 + (θ + α)CNY + α
Rab ∧ e
a ∧ eb (2.7)
where P2 =
Rab∧R
a is the Pontryagin class, CNY =
(T a∧Ta−Rab∧e
a∧eb) is the Nieh-Yan
class and we also note the Euler class E2 =
ǫabcdR
ab∧Rcd. Note that in the presence of torsion,
the action (2.7) contains the non-topological term
Rab ∧ e
a ∧ eb with “Immirzi parameter”
γ = −2/α. In the absence of torsion, this term is a total derivative.
The Hilbert-Palatini action is
IHP = IMM −
E2 . (2.8)
It differs from IMM by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)-
invariant.
– 3 –
2.1 The 3 + 1 split
Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar
from the ADM formalism, we feel it is important to set notation carefully, as we will introduce
some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will
introduce a ‘time’ function t and a foliation of space-time Σt →֒ M . In dS, t is time-like, and
this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to
be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter
σ⊥, equal to ±1 in dS(AdS).
Proceeding as usual then, we get a vector field t that satisfies ∇tt = 1 ≡ t(t) (so t =
and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as
n = σ⊥Ndt , (2.9)
which is normalized as (n, n) = σ⊥. The dual vector field n can be expanded as
N , (2.10)
where the shift N satisfies (N,n) = 0, and thus (t,n) = σ⊥N .
Next, we will locally choose a basis of 1-forms
e0 = σ⊥n = Ndt , (2.11)
eα = ẽα +Nαdt . (2.12)
The ẽα span T ∗Σt, and correspond to a 3-metric hij = ẽ
j ηαβ . The quantities N
α are the
components of N: Nα = eαi N
i. These basis 1-forms are dual to {e0 = n, eα = ẽα}, with
b) = δba.
We expand the spin connection in the same basis5
ωab = q
bdt+ ω̃
b , (2.13)
which leads to
Rab = R̃
b + dt ∧ r
b , (2.14)
where R̃ is formed from ω̃ and d̃ only, and
rab = ˙̃ω
b − d̃q
b − ω̃
b + q
b . (2.15)
Note that these quantities are merely decompositions along T ∗Σt in the 4-geometry; we will
introduce the intrinsically defined objects shortly.
We then find
IHP = 2ǫαβγ
N(R̃αβ + Λẽα ∧ ẽβ) ∧ ẽγ − 2Nα(R̃0β) ∧ ẽγ + r0α ∧ ẽβ ∧ ẽγ
. (2.16)
5We have qab = Nω0
b and ω̃
b ≡ ωα
– 4 –
As is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they
multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS4. The final
term in the action contains the real dynamics – r0α depends on the components R0α0β of the
Riemann tensor.
Note though that the tensors used here are 4-dimensional. Let us define the ”electric field”
Kα = σ⊥ω̃
α = Kβαẽ
β . (2.17)
In the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard
definition for extrinsic curvature, regarded as a vector-valued one-form. We then find
R̃αβ =
(3)Rαβ − σ⊥K
α ∧Kβ , (2.18)
R̃0α = σ⊥(d̃Kα +Kβ ∧ ω̃
α) ≡ σ⊥(D̃K)α . (2.19)
These equations amount to the Gauss-Codazzi relations.
Furthermore, r0α contains time derivatives of ω̃0α as well as terms linear in components of
q. We find
2ǫαβγr
0α ∧ ẽβ ∧ ẽγ = 2ǫαβγ
α − (D̃q)0α
∧ ẽβ ∧ ẽγ , (2.20)
= 2σ⊥ǫαβγ
K̇α + qαδKδ
∧ ẽβ ∧ ẽγ + 4q0α
ǫαβγ T̃
β ∧ ẽγ
up to a total 3-derivative. We have defined the intrinsic 3-torsion T̃ α = d̃ẽα + ω̃α β ∧ ẽ
β. Since
we wish to regard the ẽ as coordinate variables,6 we integrate the first term by parts to obtain
(up to the total time-derivative ∂
α ∧ ẽβ ∧ ẽγǫαβγ
2ǫαβγr
0α ∧ ẽβ ∧ ẽγ = Πα ∧ ˙̃e
α + 4q0αǫαβγ T̃
β ∧ ẽγ + 2σ⊥q
αδǫαβγKδ ∧ ẽ
β ∧ ẽγ . (2.21)
where we have defined the momentum 2-form
Πα = −4σ⊥ǫαβγK
β ∧ ẽγ . (2.22)
The qab appear as Lagrange multipliers. In particular, the qαβ constraint precisely sets the
antisymmetric (torsional) part of the extrinsic curvature tensor K[αβ] to zero. Next, we define
the ”magnetic field”
σ⊥ǫαβγω̃
βγ, ωαβ = −ǫαβγBγ. (2.23)
and we find that the q0α constraint
ǫαβγ T̃
β ∧ ẽγ = ǫαβγ d̃ẽ
β ∧ ẽγ − σ⊥Bβ ∧ ẽ
β ∧ ẽα = 0 , (2.24)
6Without this integration by parts, we would be in the Ashtekar formalism. Here, our choice gives a formalism
closely related to the metric variable formalism. Note that the induced boundary term may be written − 1
Πα∧ ẽ
– 5 –
involves only the antisymmetric part B[α,β] of the magnetic field Bα = Bαβ ẽ
β . The antisymmetric
part of Bα spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the
dreibein ẽα, hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3)
rotation. On the other hand, an algebraic equation of motion connects the symmetric part of
Bαβ to derivatives of ẽ
d̃ẽα + ǫαβγBβ ∧ ẽγ = 0 (2.25)
At the end, one is left with the canonically conjugate variables ẽα and Πα. These results are
familiar from the metric formalism.
Dropping the torsional terms, we then arrive at the action
IHP =
˙̃eα ∧ Πα + 2Nǫαβγ(
(3)Rαβ − σ⊥K
α ∧Kβ + Λẽα ∧ ẽβ) ∧ ẽγ
−4σ⊥N
αǫαβγ(D̃K)
β ∧ ẽγ
. (2.26)
Furthermore, using ∗3ẽ
α ∧ ẽβ = 1
αβγ ẽδ, we have
Π̂α = ∗3Πα = −2(Kαβ − ηαβtrK)ẽ
β , (2.27)
where trK = ηαβKαβ . We can solve the above equation to get
Kα = −
(Π̂αβ −
ηαβtrΠ̂)ẽ
β . (2.28)
As stated above, Kαβ (and Π̂αβ) is symmetric when the torsion vanishes.
Finally, with the definition (2.23) we find7
(3)Rαβ ∧ ẽγ = ǫαβγ
d̃ω̃αβ + ω̃αδ ∧ ω̃
∧ ẽγ = σ⊥
2d̃Bγ + ǫαβγB
α ∧ Bβ
∧ ẽγ . (2.29)
Introducing Bα is an unusual thing to do but it will play a role in duality: in this form, the
Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP
action is of the form
IHP =
˙̃eα ∧Πα − 4σ⊥N
αǫαβγ(D̃K)
β ∧ ẽγ
+2σ⊥N(2d̃Bγ + ǫαβγB
α ∧ Bβ − ǫαβγK
α ∧Kβ + σ⊥Λǫαβγ ẽ
α ∧ ẽβ) ∧ ẽγ
. (2.30)
Note that the entire contribution of the cosmological constant appears in the last term of the
Hamiltonian constraint.
7The spatial signature σ3 appears in ǫαβγǫφδρη
αφ = σ3(ηβδηγρ − ηβρηγδ). We will always consider Lorentzian
spacetime signature, so σ3 = −σ⊥.
– 6 –
2.2 Shifted Variables
It is possible to make a transformation of the canonical variables in order to absorb the cosmo-
logical constant term in (2.30). This can be achieved by introducing the new variables
K̂α = Kα − ρẽα , (2.31)
and requiring that
ρ2 = σ⊥Λ . (2.32)
This is positive only when σ⊥ and Λ are simultaneously positive or negative, as it is the case for
both AdS4 (Λ > 0) and dS4 (Λ < 0). We will often write Λ = σ⊥/L
2 where L is a length scale.
Under (2.31) the momentum 2-form becomes
Πα → Pα − 4σ⊥ρǫαβγ ẽ
β ∧ ẽγ . (2.33)
The last term in (2.33) contributes a total time derivative to the action (of the form of a boundary
cosmological term). We have introduced a new momentum variable
Pα = −4σ⊥ǫαβγK̂
β ∧ ẽγ .
Then, we get the action
IHP =
˙̃eα ∧ Pα − 4σ⊥N
αǫαβγ(D̃K̂ + ρT̃ )
β ∧ ẽγ −
σ⊥ρǫαβγ
(ẽα ∧ ẽβ ∧ ẽγ)
+2σ⊥N
2d̃(Bα ∧ ẽ
α) + 2Bγ ∧ T̃
γ − ǫαβγ
Bα ∧Bβ + K̂α ∧ K̂β + 2ρK̂α ∧ ẽβ
∧ ẽγ
.(2.34)
Note that the shift constraint is still written in terms of the ordinary covariant derivative, and
thus involves a non-linear term coupling B to K̂. Consistent with our previous discussion, we
drop the terms involving the torsion T̃ , and disregard the boundary term to obtain
IHP =
˙̃eα ∧ Pα − 4σ⊥N
αǫαβγ(D̃K̂)
β ∧ ẽγ
+2σ⊥N
2d̃(Bα ∧ ẽ
α)− ǫαβγ
Bα ∧Bβ + K̂α ∧ K̂β + 2ρK̂α ∧ ẽβ
∧ ẽγ
. (2.35)
We note that the parameter ρ can be of either sign (although, this sign does not appear in the
second order equations of motion).
2.3 Linearization
Next, we linearize the above action around an appropriate fixed background. We expand as
ẽα = ẽα + Eα, N = 1 + n, Nα = nα, Bα = Bα + bα, K̂α = K̂
+ kα . (2.36)
The background values should satisfy the constraints. The simplest choice is the background
where
= 0 = Bα . (2.37)
– 7 –
In fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order
in the fluctuating fields the Hamiltonian gives
IHP =
Ėα ∧ pα − 4σ⊥n
αǫαβγ d̃k
β ∧ ẽγ + 4σ⊥n
d̃(bα ∧ ẽ
α)− ρǫαβγk
α ∧ ẽβ ∧ ẽγ
−2σ⊥ǫαβγ
bα ∧ bβ + kα ∧ kβ + 2ρkα ∧ Eβ
∧ ẽγ
,(2.38)
where
pα = −4σ⊥ǫαβγk
β ∧ ẽγ (2.39)
are the linearized momentum variables conjugate to Eα.
In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this
to happen we find the relationships
˙̃eα + ρẽα = 0 . (2.40)
Notice that we can also write the linearized action in the form
IHP =
(Ėα + ρEα) ∧ pα − 2σ⊥ǫαβγ
bα ∧ bβ + kα ∧ kβ
∧ ẽγ
−4σ⊥n
αǫαβγ d̃k
β ∧ ẽγ + n
4σ⊥d̃bγ + ρpγ
∧ ẽγ
. (2.41)
The form of the first term, involving the momentum, makes clear that longitudinal fluctuations
are non-dynamical. The natural time dependence of Eα is of the form e−ρt (correspondingly, the
natural time dependence of pα is e
+ρt). Other than that, we see that in comparing to the flat
space action, in these variables, the only change is that the Hamiltonian constraint is modified.
The solutions of (2.40) and (2.37) are components of (A)dS4 spacetimes. We can solve (2.40)
to obtain
e0 = dt, eα = e−ρtdxα . (2.42)
With these we construct the usual Poincaré metric on (A)dS which, however, covers only half of
the space even though the parameter t runs from −∞ to +∞. The conformal boundary in these
coordinates is at t = +∞. Then we derive
ωα0 = −ρe
−ρtdxα = −ρeα , (2.43)
and so
Rαβ = −
eα ∧ eβ
Rα0 = −
eα ∧ e0
Rab = −
ea ∧ eb . (2.44)
Hence Ricab = −
ηab and R = −12σ⊥/L
2 = −12Λ. We also evaluate
Πα = −4σ⊥ρǫαβγ ẽ
β ∧ ẽγ , Π̂
= 4ρẽα, trΠ̂ = 12ρ (2.45)
Bα = 0, K
α = ρẽα ⇒ K̂
= 0 (2.46)
Note that in this gauge, (D̃K)α = 1
= 0, which solves the shift constraint, while the Hamil-
tonian constraint is satisfied through a cancellation between the K2 term and the cosmological
term.
– 8 –
3. Linearized Gravitational Duality and Holography
Let us summarize what we have obtained so far. In the presence of a cosmological constant
we have defined variables such that the action resembles most closely the action without the
cosmological constant. This was done in order to look for a suitable background around which
linear fluctuations are as simple as possible. Requiring that K̂ (the “electric field”) and B (the
”magnetic field”) vanish in such a background - as they do around flat space - we found that
the background should be (A)dS4. Quite satisfactorily, both sign choices for ρ in the change of
variables (2.31) lead to (A)dS4 spacetimes.
3.1 Duality and Holography
This is the appropriate point to recall some salient features of duality rotations. In simple
Hamiltonian systems the effect of the canonical transformation p 7→ q and q 7→ −p to the action
is (see e.g. [19])
dt[pq̇ −H(p, q)] 7→ ID =
dt[−qṗ−H(q,−p)] . (3.1)
Notice that ID involves the dual variables, for which we have however kept the same notation for
simplicity. The transformed Hamiltonian H(q,−p) is in general not related to H(p, q). However,
if H(q,−p) = H(p, q) we call the above transformation a duality. It then holds
ID = I − qp
. (3.2)
The dual action describes exactly the same dynamics as the initial one, up to a modification of
the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary,
ID is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates
the role of duality in holography; a bulk duality transformation corresponds to a particular
modification of the boundary conditions. This property of duality transformations is behind the
remarkable holographic properties of electormagnetism in (A)dS4 [2, 3].
Clearly, the crucial properties of a duality transformation are to be canonical and to leave
the Hamiltonian unchanged. However, consider a slight generalization
dt[pq̇ −
(p2 + q2 + 2λpq)] (3.3)
where λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical
transformation p 7→ q and q 7→ −p – the pq term changes sign. Consequently, the first order form
of the equations of motion are also not duality invariant. Nevertheless, the second order equation
of motion is invariant. We will find that gravity in the presence of a cosmological constant follows
precisely this model. Of course, gravity is a much more complicated constrained system, but as
we will show, the constraints and Bianchi identities transform appropriately.
– 9 –
We also note that the canonical transformation (implemented by a generating functional of
the first kind)
p 7→ q + 2λp , q 7→ −p . (3.4)
is of interest here. The above does not change the Hamiltonian and the transformed action differs
from the initial one by total time derivative terms8
S 7→ SD = S − pq
. (3.5)
3.2 Linearized gravitational duality
As a preamble to gravity we recall the duality properties of Maxwell theory
IMax =
A ∧ ∗3E −
(E ∧ ∗3E +B ∧ ∗3B)−A0d̃ ∗3 E
, (3.6)
Under the duality E 7→ − ∗3 B, B 7→ ∗3E, Ã 7→ ÃD, we find
IMax 7→ IMax,D =
AD ∧ B −
(E ∧ ∗3E +B ∧ ∗3B) + A0d̃B
. (3.7)
E and B in (3.7) should be expressed through ÃD. We observe that the kinetic term has changed
sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to
the trivial ‘Bianchi’ identity dB = 0 for the dual magnetic field.
Next we try to apply a Maxwell-type duality map in gravity. We consider the following
transformation around the fixed background (2.40)
kα 7→ −bα, bα 7→ kα . (3.8)
To implement the map (3.8) we need to specify the mapping of Eα to a ‘dual 3-bein’ Eα. We do
that using the linearized form of (2.25) as
ǫαβγbβ ∧ ẽγ + d̃E
α = 0 7→ ǫαβγkβ ∧ ẽγ + d̃E
α = 0 = d̃Eα −
pα (3.9)
Since pα = 4σ⊥d̃Eα, it is natural to define
pD,α = 4σ⊥d̃Eα = −4σ⊥ǫαβγb
β ∧ ẽγ , (3.10)
and thus the mapping (3.8) is supplemented by
E 7→ E , E 7→ −E , p 7→ −pD , pD 7→ p (3.11)
8In holography, the latter terms correspond to the relevant ”multi-trace” boundary deformations discussed in
– 10 –
Now, let us see the effects of the above duality mapping. The action transforms to
IHP 7→ IHP,D =
−Ėα ∧ pD,α − ρE
α ∧ pD,α − 2σ⊥ǫαβγ
bα ∧ bβ + kα ∧ kβ
∧ ẽγ (3.12)
+4σ⊥n
αǫαβγ d̃b
β ∧ ẽγ + n
4σ⊥d̃kγ + ρpD,α
∧ ẽγ
where now kα and bα should be expressed in terms of the dual variables Eα and pD,α via (3.9) and
(3.10). We notice that the ’kinetic’ part Ė ∧ p of the action changes sign under the duality map,
in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the
change of sign of the second term in the first line of (3.12). We will discuss this further in a later
section. For now, we note that this sign change would not show up in the equations of motion,
written in second order form. It is important to also note that the constraints are transformed
into quantities which in the next subsection we will recognize as the linearized Bianchi identities.
This is to be expected since the duality transformations are canonical. We also note that it may
be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian
invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in
the Appendix.
3.3 Linearized Constraints and Bianchi Identities
By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8)
the linearized constraints transform to the linearized Bianchi identities as
Cα ≡ ǫαβγ d̃k
β ∧ ẽγ 7→ −ǫαβγ d̃b
β ∧ ẽγ (3.13)
C0 ≡ −σ⊥
d̃bγ − ρǫαβγk
α ∧ ẽβ
∧ ẽγ 7→ −σ⊥
d̃kγ + ρǫαβγb
α ∧ ẽβ
∧ ẽγ (3.14)
To identify the right hand sides, we first note that the Bianchi identities are
b = dR
c ∧ ω
b + ω
b = 0 (3.15)
BaT = dT
a − Rab ∧ e
b + ωab ∧ T
b = 0 (3.16)
which are obtained from the definitions of Rab and T
a by exterior differentiation. The first
equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that
Rab ∧ e
b = 0. If we do the 3+1 split, we find two equations. The first is
α = −((3)Rαβ − σ⊥K
α ∧Kβ) ∧ ẽ
β = 0 (3.17)
which upon using the symmetry of Kα linearizes to
α = −ǫαβγ d̃b
β ∧ ẽγ + . . . (3.18)
Note that this is the image under duality of the shift constraint as in (3.13).
The second identity is
0 = −R̃0α ∧ ẽ
α = −σ⊥(D̃K)α
= −σ⊥
d̃kα + ρǫαβγb
β ∧ ẽγ
∧ ẽα = 0 (3.19)
– 11 –
where to arrive in the second line we used (2.46). This is the image of the Hamiltonian constraint
as in (3.14).
Summarizing, the duality transformations between linearized constraints and Bianchi iden-
tities are
Cα 7→ BT,α C0 7→ B
T (3.20)
BT,α 7→ −Cα B
T 7→ −C0 (3.21)
3.4 Connection with other known dualities
The Maxwell-type duality operation (3.8) is closely related to the dualization of the first two
indices of the Riemann tensor as9
Rab → S
dRcd (3.22)
at least at the linearized level. Let us investigate (3.22) by rewriting expressions in the 3+1 split.
We have
Rab = R̃
b + dt ∧ r
Sab = S̃
b + dt ∧ s
We begin with the spatial 2-forms when we have
R̃αβ = −ǫαβγ d̃Bγ + σ⊥(B
α ∧ Bβ −Kα ∧Kβ) (3.23)
R̃0α = σ⊥(d̃Kα +Kβ ∧ ω̃
α) ≡ σ⊥(D̃K)α (3.24)
S̃0γ =
σ⊥ǫαβγR̃
αβ (3.25)
S̃αβ = ǫαβγR̃
0γ (3.26)
If we linearize these expressions, we find under the duality transformation (3.8)
R̃ab 7→ −σ⊥S̃
ab (3.27)
Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an ‘integrated
form’ of the usual Riemann tensor duality, but implies it.
Similarly, if we investigate the spatial 1-forms, we find
rab 7→ −σ⊥s
ab (3.28)
To arrive at this result we have set to zero the Lagrange multiplier field q.
9For a discussion of the duality properties of gravity in terms of the Riemann tensor see [10].
– 12 –
4. The Effect on the Boundary Theory
It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how
the duality transformation that we have defined here acts in the boundary. We are instructed
to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a
solution to the equation of motion, resulting in a pure boundary term which is of the form
Sbdy =
pα ∧ E
α (4.1)
Applying the duality transformation to the bulk theory, although the bulk action is not invari-
ant as we have discussed above, nevertheless it may be easily shown that it induces a simple
transformation on the (linearized) boundary term: it simply changes its sign.
Sdualbdy = −
pD,α ∧ E
α (4.2)
This transformation is exactly analogous to what happens in the Maxwell case: it amounts to
the result [21].10
2 = −1 . (4.3)
5. Conclusions and Outlook
Motivated by possible application in holography and in higher-spin gauge theory we have studied
the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity
action in terms of suitable variables that closely resemble the electric and magnetic fields in
Maxwell theory. We have found suitable ”electric” and ”magnetic” field variables, such that at
the linearized level first order gravity most closely resembles electromagnetism. This can be done
only around Minkowksi and (A)dS4 backgrounds.
We have implemented duality transformations in the linearized gravity fluctuations around
these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nev-
ertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints
are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce
boundary terms whose relevance in holography we have briefly discussed. Finally, we have ex-
hibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces
boundary terms corresponding to relevant deformations.
The main implication of our results is that certain properties of correlations functions in
three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our
results to black-hole backgrounds and also when topological terms are present in the bulk. We
also expect that one can analyze the duality of higher-spin gauge theories based on our first-order
approach.
Acknowledgments
10See also [22] for an interesting recent application of this formula.
– 13 –
The work of A. C. P. was partially supported by the research program ”PYTHAGORAS
II” of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of
Energy under contract DE-FG02-91ER40709.
6. Appendix: other duality mappings
It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the
following transformation in the fixed background (2.40)
kα 7→ −bα − 2ρEα, bα 7→ kα . (6.1)
The mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals
that the action transforms as
IHP 7→ IHP,D = IHP + 4σ⊥
ǫαβγE
α ∧ bβ + ρǫαβγE
α ∧ Eβ
∧ ẽγ
αǫαβγ d̃b
β ∧ ẽγ − 8ρnαkβ ∧ ẽα ∧ ẽ
+n(4σ⊥d̃kα + 4σ⊥ρǫαβγb
β ∧ ẽγ + 8ΛǫαβγE
β ∧ ẽγ) ∧ ẽα
(6.2)
The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total
”time” derivative terms shown in the first line of (6.2).
Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us
see that in some detail. The second term in the shift constraint is zero since kα is a symmetric
one form kα = kαβ ẽ
β with kαβ = kβα; see (2.21).
The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved
to see and it is based on the possibility of solving (3.9) for Eα after gauge fixing.11 One way to
see this is in components. Write Eα = Eαβ ẽ
β and (3.9) becomes
γ − ∂γE
α = ǫ
γ − ǫ
α (6.3)
In the ”Lorentz gauge” where ∂αEβα = 0 = ∂
αkβα the above can be inverted as
Eαβ =
ǫαδγ∂
γkδβ (6.4)
Using (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality
transformation is probably related to the one considered by Julia et. al. in [15].
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