#### Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry

HJE
Hoˇrava-Lifshitz gravity from dynamical Newton-Cartan geometry
Jelle Hartong 0 1 2 4
Niels A. Obers 0 1 2 3
0 Blegdamsvej 17 , DK-2100 Copenhagen Ø , Denmark
1 C. P. 231, 1050 Brussels , Belgium
2 Universit ́e Libre de Bruxelles
3 The Niels Bohr Institute, Copenhagen University
4 Physique Th ́eorique et Math ́ematique and International Solvay Institutes
Recently it has been established that torsional Newton-Cartan (TNC) geometry is the appropriate geometrical framework to which non-relativistic field theories couple. We show that when these geometries are made dynamical they give rise to HoˇravaLifshitz (HL) gravity. Projectable HL gravity corresponds to dynamical Newton-Cartan (NC) geometry without torsion and non-projectable HL gravity corresponds to dynamical NC geometry with twistless torsion (hypersurface orthogonal foliation). We build a precise dictionary relating all fields (including the scalar khronon), their transformations and other properties in both HL gravity and dynamical TNC geometry. We use TNC invariance to construct the effective action for dynamical twistless torsional Newton-Cartan geometries in 2+1 dimensions for dynamical exponent 1 < z ≤ 2 and demonstrate that this exactly agrees with the most general forms of the HL actions constructed in the literature. Further, we identify the origin of the U(1) symmetry observed by Hoˇrava and Melby-Thompson as coming from the Bargmann extension of the local Galilean algebra that acts on the tangent space to TNC geometries. We argue that TNC geometry, which is manifestly diffeomorphism covariant, is a natural geometrical framework underlying HL gravity and discuss some of its implications.
Classical Theories of Gravity; Space-Time Symmetries; Models of Quantum
Gravity
1 Introduction
2 Local Galilean transformations
3 The affine connection: part 1
4 Local Bargmann transformations
5 The affine connection: part 2
6 Torsion and the Stu¨ckelberg scalar
7 Curvatures
8 Coordinate (ADM) parametrizations
9 Hoˇrava-Lifshitz actions
10 Local Bargmann invariance of the HL action: local U(
1
)
vs. Stu¨ckelberg coupling
11 A constraint equation
12 Conformal HL gravity from the Schro¨dinger algebra
13 Discussion
A Gauging Poincar´e
1
Introduction
completion of gravity. While observational constraints and the matching to general
relativity in the IR put severe limitations on the phenomenological viability of this proposal, HL
gravity is of intrinsic theoretical interest as an example of gravity with anisotropic scaling
between time and space. In particular, in the context of holography it holds the prospect
of providing an alternative way [3, 4] of constructing gravity duals for strongly coupled
systems with non-relativistic scaling, including those of interest to condensed matter physics.
More generally, one might expect that HL gravity has a natural embedding in the larger
framework of string theory [5].
In parallel to this development, and with in part similar motivations, there has been
considerable effort to extend the original AdS-setup in (conventional) relativistic gravity
to space-times with non-relativistic scaling [6–9]. Such space-times typically exhibit a
dynamical exponent z that characterizes the anisotropy between time and space on the
– 1 –
boundary. This includes in particular holography for Lifshitz space-times, for which it was
found that the boundary geometry is described by a novel extension of Newton-Cartan (NC)
geometry1 with a specific torsion tensor, called torsional Newton-Cartan (TNC) geometry.
The aim of this paper is to construct the theory of dynamical TNC geometry and show
that it exactly agrees with the most general forms of HL gravity.
TNC geometry was first observed in [17, 18] as the boundary geometry for a specific
action supporting z = 2 Lifshitz geometries, and subsequently generalized to a large class
of holographic Lifshitz models for arbitrary values of z in [19, 20]. In parallel, it was shown
in detail in [21] how TNC geometry arises by gauging the Schr¨odinger algebra, following
the earlier work [22] on obtaining NC geometry from gauging the Bargmann algebra. In
metric hμν and a vector field Mμ = mμ − ∂μχ where χ is a Stu¨ckelberg scalar whose role
in TNC geometry will be elucidated in section 6. The torsion in TNC geometry is always
depends on the properties of τμ and we distinguish the three cases:3
proportional to ∂μτν − ∂ν τμ where τμ defines the local flow of time. The amount of torsion
• Newton-Cartan (NC) geometry
• twistless torsional (TTNC) geometry
• torsional Newton-Cartan (TNC) geometry
1We refer to [10–16] for earlier work on Newton-Cartan geometry.
2Ref. [25] introduced NC geometry to field theory analyses of problems with strongly correlated electrons,
such as the fractional quantum Hall effect. Later torsion was added to this analysis in [26]. The type of
torsion introduced there is what we call twistless torsion. See also [27–29] for a different approach to
Newton-Cartan geometry.
3These three cases also naturally arise in Lifshitz holography [17, 18]. We note that TTNC geometry
was already observed in [16] but in that work the torsion was eliminated using a conformal rescaling.
– 2 –
where the first possibility has no torsion and the latter option has general torsion with the
twistless case being an important in-between situation. More specifically, in the first case
the time-like vielbein of the geometry is closed and defines an absolute time. In the second
case the time-like vielbein is hypersurface orthogonal and thereby allows for a foliation of
equal time spatial surfaces described by Riemannian (i.e. torsion free) geometry. In the
third, most general, case there is no constraint on τμ.
As is clear from holographic studies of the boundary energy-momentum tensor as for
example in [17–19, 23, 31] the addition of torsion to the NC geometry is crucial in order to
be able to calculate the energy density and energy flux of the theory. This is because they
are the response to varying τμ (see also [30]). Hence in order to be able to compute these
quantities τμ better be unconstrained, i.e. one should allow for arbitrary torsion. If we work
with TTNC geometry one can only compute the energy density and the divergence of the
energy current [20] because in that case τμ = ψ∂μτ where one has to vary ψ and τ with ψ
sourcing the energy density and τ sourcing the divergence (after partial integration) of the
energy current. In any case the point is that, contrary to the relativistic setting, adding
torsion is a very natural thing to do in NC geometry. Moreover, as will be shown later,
the torsion is not something one can freely pick and is actually fixed by the formalism.
In all of these works the TNC geometry appears as a fixed background and is hence
not dynamical. The purpose of this paper is to consider what theory of gravity appears
when letting the TNC geometry fluctuate. We find, perhaps not entirely unexpected,4 that
depending on the amount of torsion the resulting theories include HL gravity and all of its
known extensions.
Our focus in this paper will be mainly on the first two of the three cases listed above,
leaving the details of the dynamics of the most general case (TNC gravity) for future work.
In particular, we will show that:
• dynamical NC geometry = projectable HL gravity
• dynamical TTNC geometry = non-projectable HL gravity.
The khronon field introduced by [32] (to make HL gravity generally covariant whereby
making manifest the presence of an extra scalar mode) naturally appears (see also [33]) in
our formulation. We furthermore show that the U(
1
) extension of [34] (see also [35–37])
emerges as well in a natural fashion. The essential identification between the covariant5
NC-type geometric structures and those appearing in the ADM parametrization that forms
the starting point of HL gravity is as follows
τμ ∼ lapse ,
hˆμν ∼ spatial metric ,
mμ ∼ shift + Newtonian potential ,
where the fields hˆμν and mμ are defined in section 4. We will show that the effective action
for the TTNC fields leads to two kinetic terms for the metric hˆμν (giving rise to the λ
parameter of HL gravity [1, 2]) including the potential terms computed in refs. [32, 36, 37, 39].
4A HL-type action in TNC covariant form was already observed in [18] where the anisotropic
Weylanomaly in a specific z = 2 holographic four-dimensional bulk Lifshitz model was obtained via null
ScherkSchwarz reduction of the AdS5 conformal anomaly of gravity coupled to an axion.
5Note that in e.g. ref. [38] there is also a type of covariantization of HL gravity (see also eq. (3.9) of [3]),
but there is still inherently a Lorentzian metric structure present. This only works up to second order in
derivatives so that it only captures the IR limit of HL gravity.
– 3 –
Furthermore the Stu¨ckelberg scalar χ entering in the TNC quantity Mμ = mμ − ∂μχ
(see [17–19, 21, 23, 24]) will be directly related to the Newtonian prepotential introduced
in [34]. The relation to TTNC geometry will, however, provide a new perspective on the
nature of the U(
1
) symmetry studied in the context of HL gravity. As a further confirmation
that TNC geometry is a natural framework for HL gravity we will demonstrate in this
paper that when we include dilatation symmetry (local Schro¨dinger invariance) one obtains
conformal HL gravity.
As we will review in this paper, the various versions of TNC geometry defined above
arise by gauging non-relativistic symmetry algebras (Galilean, Bargmann, Schr¨odinger).
In particular, in this procedure the internal symmetries are made into local symmetries,
and translations are turned into diffeomorphisms. This is in the same way that
Riemannian geometry comes from gauging the Poincar´e algebra, thereby imposing local Lorentz
symmetry and turning translations into space-time diffeomorphisms. Thus HL gravity
theories (and more generally TNC gravity) can be seen as the most general gravity theories
for which the Einstein equivalence principle (that locally space-time is described by flat
Minkowski space-time) is applied to local non-relativistic (Galilean) symmetries, rather
than to the local Lorentz symmetry that one has in special relativity.
We point out that in general relativity (GR) the global symmetries (Killing vectors)
of Minkowski space-time (the Poincar´e algebra) form the same algebra from which upon
gauging (and replacing local space-time translations by diffeomorphisms as explained in
appendix A) we obtain the geometrical framework of GR. On the other hand the Killing
vectors of flat NC space-time only involve space and time translations and spatial
rotations [23] while the local tangent space group that we gauge in order to obtain the TNC
geometrical framework is the Galilean algebra (where again we also replace local time and
space translations by diffeomorphisms), which also contains Galilean boosts and is thus not
the same algebra as the algebra of Killing vectors of flat NC space-time. We bring this up
to highlight the fact that the local tangent space symmetries and the Killing vectors of flat
space-time are in general two very different concepts that are often mistakenly assumed
to be the same. Basically this happens because the Mμ vector allows for the construction
of a new set of vielbeins (defined in section 4) that are invariant under G transformations
and that only see diffeomorphisms and local rotations which agrees with the Killing vectors
of flat NC space-time. Nevertheless the fact that Mμ is one of the background fields to
which we can couple a field theory can, under special circumstances, lead to additional
symmetries such as G and N (and even special conformal symmetries) [23].
Our results on dynamical TNC geometry and its relation to HL gravity provide a new
perspective on these theories of gravity. For one thing, the vacuum of HL gravity (without
a cosmological constant) has so far been taken to be Minkowski space-time, but since the
underlying geometry appears to be TNC geometry, it seems more natural to take this as flat
NC space-time [23, 24]. Thus it would seem worthwhile to reexamine HL gravity and the
various issues6 that have been raised following its introduction. As another application,
we emphasize that, independent of a possible UV completion of gravity, our results on
dynamical TNC geometry are of relevance to constructing IR effective field theories of
6There is an extensive literature on this (e.g. instabilities and strong coupling at low energies), see e.g.
refs. [40–47].
– 4 –
non-relativistic systems following the recent developments of applying this to condensed
matter systems. For these kinds of applications, the question whether HL gravity flows
to a theory with local Lorentz invariance (λ = 1) in the IR is of no concern. Finally,
from a broader perspective our results might be useful towards a proper description of
the non-relativistic quantum gravity corner of the “(~, GN , 1/c)-cube”, perhaps aiding the
formulation of a well-defined perturbative 1/c expansion around such a theory.
Outline of the paper.
The first part of the paper (sections 2 to 7) is devoted to setting
up the geometrical framework for torsional Newton-Cartan geometry, presented in such a
way that the subsequent connection to HL gravity is most clearly displayed. We thus take
a pedagogical approach that introduces the relevant ingredients in a step-by-step way. To
this end we begin in section 2 with the geometry that is obtained by gauging the Galilean
algebra, extending the original work of [22] to include torsion. We exhibit the
transformation properties of the relevant geometrical fields under space-time diffeomorphisms and
the internal transformations, consisting of Galilean boosts (G) and spatial rotations (J ).
We also discuss the vielbein postulates and curvatures entering the field strength of the
gauge field. We point out that the only G, J invariants are the time-like vielbein τμ and the
inverse spatial metric hμν . In section 3 we then present the most general affine connection
that satisfies the property that the latter quantities are covariantly conserved.
In section 4, we go one step further and add the central element (N ) to the Galilean
algebra, and consider the gauging of the resulting Bargmann algebra (as also considered
in [22] for the case with no torsion). We show that the extra gauge field mμ that enters
in this description, does not alter the transformation properties of the objects considered
in section 2, but allows for the introduction of further useful G, J, invariants, namely an
inverse time-like vielbein vˆμ, a spatial metric h¯μν (or hˆμν ) and a “Newtonian potential” Φ˜.
We then return to the construction of the affine connection in section 5 and employ the
geometric quantities of section 2 and 4 to construct the most general connection that can
be built out of the invariants. We discuss two special choices of affine connections with
particular properties, one of them being especially convenient for the comparison with
HL gravity. We point out that, in the case of non-vanishing torsion, there is no choice
of affine connection that is also N -invariant, but that one can formally remedy this by
introducing a Stu¨ckelberg scalar χ (defining Mμ = mμ − ∂μχ) to the setup that cancels
this non-invariance. This has the advantage that one can deal simultaneously with theories
that have a local U(
1
) symmetry and those that do not have this, and further it will prove
useful when comparing to HL gravity (especially [34–37]). We also show how the TNC
invariants can be used to build a non-degenerate symmetric rank 2 tensor with Lorentzian
signature, which will later be used to make contact with the ADM decomposition that
enters HL gravity.
In section 6 we discuss the specific form of the torsion tensor that emerges from gauging
the Bargmann algebra and introduce the three relevant cases for torsion (NC, TTNC and
TNC) that were already mentioned above. We also introduce a vector aμ that describes
the TTNC torsion, which will turn out to be very useful in order to make contact with the
literature on non-projectable HL gravity. Further we will identify the khronon field of [32].
– 5 –
Then in section 7 we give some basic properties of the curvatures (extrinsic curvature and
Ricci tensor for TTNC) that will be useful when constructing HL actions.
In section 8 we relate the TNC invariants introduced in the previous sections to those
appearing in the corresponding ADM parameterization employed in HL gravity. This
identification and the match of the properties and number of components and local symmetries
in the case of NC and TTNC already strongly suggest that dynamical (TT)NC is expected
to be the same as (non)-projectable HL gravity. We then proceed in section 9 by showing
that the generic action that describes dynamical TTNC geometries agrees on the nose with
the most general HL actions appearing in the literature. For simplicity we treat the case
of 2 spatial dimensions with 1 < z ≤ 2 and organize the terms in the action according
to their dilatation weight. In particular, we construct all G, J invariant terms that are
relevant or marginal, using as building blocks the TNC invariants (including the torsion
tensor and curvature tensor) and covariant derivatives. The resulting action is written
in (9.18), (9.19) and gives the HL kinetic terms [1, 2] while the potential is exactly the
same as the 3D version of the potential given in [32, 36, 37, 39].
We then proceed in section 10 to consider the extension of the action to include
invariance under the central extension N , leading to HL actions with local Bargmann invariance.
This can be achieved by including couplings to Φ˜, which did not appear yet in section 9.
Importantly, in the projectable case with the HL coupling constant λ = 1 we reproduce
the U(
1
) invariant action of [34]. When we consider the non-projectable version or λ 6= 1
we need additional terms to make the theory U(
1
) invariant which is precisely achieved by
adding the Stu¨ckelberg field χ that we introduced in section 5 (see also [23]). We can then
write a Bargmann invariant action that precisely reproduces the actions considered in the
literature, where in particular the χ-dependent pieces agree with those in [36, 37]. This
comes about in part via coupling to the natural TNC Newton potential, Φ˜ χ, which is the
Bargmann invariant generalization of Φ˜ , and the simple covariant form of the action (10.14)
is one of our central results.
We emphasize that adding the χ field to the action means that we have trivialized
the U(
1
) symmetry by Stu¨ckelberging it or in other words we have removed the U(
1
)
transformations all together. We further expand on this fact in section 11, commenting
on statements in the literature regarding the relevance of the U(
1
) invariance (which is
not there unless we have zero torsion and λ = 1) in relation to the elimination of a scalar
degree of freedom. In particular, we will present a different mechanism that accomplishes
this and which involves a constraint equation obtained by varying the TNC potential Φ˜χ.
Finally in section 12 we consider the case where we add dilatations to the Bargmann
algebra, i.e. we consider the dynamics we get from a geometry that is locally Schr¨odinger
invariant. We will show that the resulting theory is conformal HL gravity, providing further
evidence for our claim that TNC geometry is the underlying geometry of HL gravity. In
particular, employing the local Schr¨odinger algebra we will arrive at the invariant z = d
action (12.50) for conformal HL gravity in d + 1 dimensions.
We end in section 13 with our conclusions and discuss a large variety of possible open
directions. For comparison to general relativity and as an introduction to the logic followed
in sections 2 to 7, we have included appendix A which discusses the gauging of the Poincar´e
algebra leading to Riemannian geometry (possibly with torsion added).
– 6 –
Local Galilean transformations
The present section until section 7 is devoted to setting up the general geometrical
framework for torsional Newton-Cartan geometry. We will follow an approach that is very similar
to what in general relativity is known as the gauging of the Poincar´e algebra. This provides
us in a very efficient manner with all basic geometrical objects used in the formulation of
general relativity (and higher curvature modifications thereof). For the interested reader
unfamiliar with this method we give a short summary of it in appendix A.
To obtain torsional Newton-Cartan geometry we follow the same logic as in appendix A
for the case of the Galilean algebra and its central extension known as the Bargmann
algebra. This was first considered in [22] for the case without torsion. Here we generalize
this interesting work to the case with torsion. Adding torsion to Newton-Cartan geometry
can also be done by making it locally scale invariant, i.e. gauging the Schr¨odinger algebra as
in [21]. However upon gauging the Schr¨odinger algebra the resulting geometric objects are
all dilatation covariant which is useful for the construction of conformal HL gravity as we
will study in section 12 but it is less useful for the study of general non-conformally invariant
HL actions which is why we start our analysis by adding torsion to the analysis of [22].
Consider the Galilean algebra whose generators are denoted by H, Pa, Ga, Jab and
whose commutation relations are
[H , Ga] = Pa ,
[Jab , Pc] = δacPb − δbcPa ,
[Jab , Jcd] = δacJbd − δadJbc − δbcJad + δbdJac .
[Pa , Gb] = 0 ,
Let us consider a connection Aμ taking values in the Galilean algebra7
Aμ = Hτμ + Paeaμ + GaΩμa +
1
2 JabΩμab .
This connection transforms in the adjoint as
With this transformation we can associate another transformation denoted by δ¯ as follows.
Write (without loss of generality)
where
indices.
is chosen to only include the internal symmetries G and J . We define δ¯Aμ as
δ¯Aμ = δAμ − ξν Fμν = LξAμ + ∂μΣ + [Aμ , Σ] ,
7Our notation is such that μ, ν = 0 . . . d are spacetime indices and a, b = 1 . . . d are spatial tangent space
δAμ = ∂μΛ + [Aμ , Λ] .
Λ = ξμAμ + Σ ,
2 Jabλab ,
can only be obtained once specific curvature constraints are imposed.8 We emphasize that
the transformation δ¯Aμ exists no matter what we choose for the curvature Fμν .
If we write in components what (2.6) states we obtain the transformation properties
HJEP07(215)
δ¯τμ = Lξτμ ,
δ¯eaμ = Lξeaμ + λabebμ + λaτμ ,
δ¯Ωμa = LξΩμa + ∂μλa + λabΩμb + λbΩμba ,
δ¯Ωμab = LξΩμab + ∂μλab + 2λ[acΩμ|c|b] ,
where Lξ is the Lie derivative along ξμ and λa, λab the parameters of the internal G, J
transformations, respectively.
We can now write down covariant derivatives that transform covariantly under these
transformations. They are
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
Dμτν = ∂μτν − Γρμν τρ ,
Dμeνa = ∂μeν − Γρμν eρ − Ωμaτν − Ωμabebν ,
a a
where Γρμν is an affine connection transforming as
δ¯Γρμν = ∂μ∂ν ξρ + ξσ∂σΓρμν + Γρσν ∂μξσ + Γρμσ∂ν ξσ − Γσμν ∂σξρ .
It is in particular inert under the G and J transformations. The form of the covariant
derivatives is completely fixed by the local transformations δ¯Aμ.
redefinition of the connections Γρμν , Ωμa and Ωμab that leaves the covariant derivatives
However any tensor form-invariant leads to an allowed set of connections with the exact same transformation properties.
We impose the vielbein postulates
which allows us to express Γρμν in terms of Ωμa and Ωμab via
Γμν = −vρ∂μτν + eρa ∂μeν − Ωμaτν − Ωμabebν ,
ρ a
8This is because setting to zero some of the curvatures in Fμν identifies δ¯ with δ in (2.6) for those fields
that are not fixed by the curvature constraints. There is no need for the δ and δ¯ transformations to coincide.
As we show in appendix A this is no longer the case in GR when there is non-vanishing torsion.
Dμτν = 0 ,
Dμeνa = 0 ,
– 8 –
where we defined inverse vielbeins vμ and eaμ via
The vielbein postulates for the inverses read
μ
v τμ = −1 ,
vμeaμ = 0 ,
μ
Using that Ωμab is antisymmetric we find that hμν = δabeaμebν satisfies
In other words they are equal to the torsion tensor, i.e.
The other two curvature tensors can be found by computing the Riemann tensor defined as
Using that
together with (2.17) tells us that
which together with equations (2.12) and (2.15), i.e.
constrain Γρμν . Equations (2.21) and (2.22) are the TNC analogue of metric
compatibility in GR.
The components of the field strength Fμν in (2.7) are given by
Rμν (H) = 2∂[μτν] ,
Rμν a(P ) = 2∂[μeν] − 2Ω[μaτν] − 2Ω[μabebν] ,
a
Rμν a(G) = 2∂[μΩν]a − 2Ω[μabΩν]b ,
Rμν ab(J ) = 2∂[μΩν]ab
− 2Ω[μcaΩν]bc .
The first two appear in the antisymmetric part of the covariant derivatives Dμτν and Dμeνa.
More precisely we have
∇μhνρ = 0 ,
∇μτν = 0 ,
Rμν (H) = 2Γ[ρμν]τρ ,
Rμν a(P ) = 2Γ[ρμν]eρ .
a
2Γ[ρμν] = −vρRμν (H) + eρaRμν a(P ) .
[∇μ , ∇ν ]Xσ = RμνσρXρ − 2Γ[ρμν]∇ρXσ .
Rμνσρ = −∂μΓνρσ + ∂ν Γρμσ − ΓρμλΓνλσ + ΓνρλΓλμσ ,
ρ
Rμνσρ = eρaτσRμν a(G) − eσaeb Rμν ab(J ) .
– 9 –
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
ing (2.21) and (2.22)) are all independent. The inverse vielbeins vμ and eaμ transform as
So far all components of Aμ are independent or what is the same τμ, eaμ and Γρμν
(obeyδ¯vμ = Lξvμ + eaμλa ,
δ¯eaμ = Lξeaμ + λabebμ .
There are only two invariants, i.e. tensors invariant under G and J transformations, that
we can build out of the vielbeins. These are τμ and hμν = δabeaμebν . This is not enough to
construct an affine connection that transforms as (2.14). The reason we cannot build any
other invariants is because vμ and hμν = δabeaμebν undergo shift transformations under local
Galilean boosts λa (also known as Milne boosts [30]).
where hρσYσμν satisfies
It follows that Yσμν can be written as
hλσhρν + hρσhλν Yσμν = 0 .
Yσμν = τσXμ1ν + τν Xσ2μ + Xσ3μν ,
that Kσμ = −Kμσ. Further we write Xσ3μν = τμKσν + X˜σ3μν so that we can write
where Xμ1ν and Xσ2μ and Xσ3μν = −Xν3μσ are arbitrary. We write Xσ2μ = Kσμ + X(2σμ) so
Yσμν = τσ Xμ1ν + X(2μν)
+ τμKσν + τν Kσμ + Lσμν ,
where Lσμν = −Lνμσ is defined as
Lσμν = τν X(2σμ) − τσX(2νμ) + X˜σ3μν .
Since Yσμν is defined as hρσYσμν we can drop the part in (3.4) that is proportional to τσ.
We thus find the following form for the connection Γμν
3
The affine connection: part 1
The most general Γρμν obeying (2.21) and (2.22) is of the form
Γμν = −vρ∂μτν +
ρ
2
1 hρσ (∂μhνσ + ∂ν hμσ − ∂σhμν ) +
2
1 hρσYσμν
2
Γμν = −vρ∂μτν +
ρ
2
1 hρσ (∂μhνσ + ∂ν hμσ − ∂σhμν ) +
The variation of Γρμν under local Galilean boosts yields
ρ
2
1 hρσ (τμKσν + τν Kσμ + Lσμν ) . (3.6)
2
+
2
δGΓρμν =
1 hρστμ (δGKσν + ∂ν λσ − ∂σλν ) +
1 hρστν (δKσμ + ∂μλσ − ∂σλμ)
1 hρσ (δGLσμν − λσ (∂μτν − ∂ν τμ) + λμ (∂ν τσ − ∂στν ) + λν (∂μτσ − ∂στμ)) ,
ρ
where λμ = λaeaμ. In section 5 we will choose Kμν and Lσμν such that δGΓμν = 0.
Local Bargmann transformations
It is well known that the Galilean algebra admits a central extension with central element
N called the Bargmann algebra. This latter element appears via the commutator [Pa, Gb] =
δabN . We denote the associated gauge connection by mμ. Following the same recipe as in
section 2 with
where δ¯ is defined in the same way as in (2.6). Note that we have an extra parameter σ
associated with the N transformation. Because N is central, all results of the previous
section remain unaffected.
Our primary focus in this section is local Galilean boost invariance. The new field mμ
is shifted under the λa transformation and so in combinations such as
vˆμ = vμ − hμν mν ,
h¯μν = hμν − τμmν − τν mμ ,
the Galilean boost parameter λa is cancelled. However we now have two other things to
worry about. First of all the new field mμ also transforms under a local U(
1
) transformation
with parameter σ and secondly we have introduced more than is strictly necessary to have
local Galilean invariance. This is because the component
Φ˜ = −vμmμ + 12 hμν mμmν
is G invariant (and of course also J invariant). In previous works we have introduced
another background field χ, a Stu¨ckelberg scalar, transforming as δ¯χ = Lξχ + σ so that the
combination Mμ = mμ − ∂μχ is invariant under the local N transformation and replaced
everywhere mμ by Mμ. Here it will prove convenient, for the sake of comparison with work
on HL gravity to postpone this step until later.9 Hence for now we will work with mμ as
opposed to Mμ.
and eaμ where eˆaμ = eaμ − maτμ with ma = eμamμ. They satisfy the relations
We introduce a new set of Galilean invariant vielbeins: τμ, eˆaμ whose inverses are vˆμ
μ
vˆ τμ = −1 ,
vˆμeˆaμ = 0 ,
μ
9In previous work [19, 21, 23, 24] we denoted by vˆμ, h¯μν and Φ˜ the invariants with mμ replaced by Mμ.
Here we temporarily work with the forms (4.5)–(4.7) for reasons that will become clear as we go on. We
return to our notation from previous works in section 12. We also point out that compared to [19, 21, 23, 24]
we denote by mμ here what was referred to as m˜μ in these papers and vice versa we denote by m˜μ here
what was denoted by mμ in these respective works.
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
We also have the completeness relation eaμeˆνa = δνμ + vˆμτν. The introduction of ma thus
leads to the G, J invariants vˆμ and
where h¯μν is given in (4.6). The part of mμ that is responsible for the Galilean boost
invariance is ma that transforms as (ignoring the σ transformation)
hˆμν = δabeˆaμeˆbν = h¯μν + 2τμτν Φ˜ ,
δ¯ma = Lξma + λa + λabmb .
mμ = eaμma − 2
1
mamaτμ + Φ˜ τμ ,
We can write
where the last term is an invariant.
5
The affine connection: part 2
In section 2 we realized the Galilean algebra on the fields τμ, eaμ, Ωμa and Ωμab or what
ρ
is the same on τμ, eaμ and Γμν where the affine connection obeys (2.21) and (2.22). Now
that we have introduced a new field mμ transforming as in (4.4) we will see that we can
realize the Galilean algebra on a smaller set of fields, namely τμ, eaμ and mμ. We can also
realize the Galilean algebra on τμ, eaμ and ma with ma transforming as in (4.10), i.e. no
dependence on Φ˜ or realize it on τμ, eaμ, ma and Φ˜ which is another way of writing the
dependence on τμ, eaμ and mμ. These different options lead to different choices for the affine
connection as we will now discuss.
The most straightforward way of constructing a Γρμν that is made out of vielbeins and
either i). mμ or ii). ma, that obeys (2.21) and (2.22) and transforms as in (2.14), is to use
the invariants τμ, h¯μν, vˆμ, hμν and Φ˜ . The most general connection we can build out of
these invariants reads [23]
2
2
(4.9)
(4.10)
(4.11)
where Hμν is given by
Γρμν = −vˆρ∂μτν + 1 hρσ (∂μHνσ + ∂νHμσ − ∂σHμν) ,
Hμν = h¯μν + ατμτνΦ˜ ,
Γρμν by Γ¯ρμν which is given by
where α is any constant. If we want the connection to depend linearly on mμ, which is
a special case of case i). above, we should take α = 0. If we wish that the connection is
independent of Φ˜ as in case ii). we should take α = 2 because of the identity (4.9) so that
Hμν = hˆμν where hˆμν only depends on ma. For the general case i). i.e. general dependence
on ma and Φ˜, we can take any α. For case i). with a linear dependence on mμ we denote
Γ¯ρμν = −vˆρ∂μτν + 1 hρσ ∂μh¯νσ + ∂ν h¯μσ − ∂σh¯μν .
This form of Γρμν has been used in [19, 21, 24, 30, 48]. The form of Γ¯ρμν corresponds to
taking in (3.6) the following choices for Kμν and Lσμν, namely
Kμν = ∂μmν − ∂νmμ ,
Lσμν = mσ (∂μτν − ∂ντμ) − mμ (∂ντσ − ∂στν) − mν (∂μτσ − ∂στμ) .
1 hρσ ∂μhˆνσ + ∂ν hˆμσ − ∂σhˆμν .
The two connections Γˆρμν and Γ¯ρμν differ by a tensor as follows from
Γˆρμν = Γ¯ρμν + Φ˜hρστν (∂μτσ − ∂στμ) + Φ˜ hρστμ (∂ν τσ − ∂στν ) − τμτν hρσ∂σ Φ˜ .
In this work it will prove most convenient to use the connection (5.6) as this eases
comparison with HL gravity. We stress though that in principle one can take any of the above
choices, i.e. any value for α, and that the final form of the effective action for HL gravity
will take the same form regardless which Γρμν one chooses as all dependence on α drops out
when forming the scalar terms appearing in the action.10
The reader familiar with the literature on NC geometry without torsion might wonder
which of these connections relates to the one of NC geometry (as written for example in [22]
and references therein). The usual NC connection is obtained by taking (5.3) with Kμν as
given in (5.5) and Lσμν = 0 which follows from (5.5) and the fact that for NC geometry we
have ∂μτν −∂ν τμ = 0. The possibility of modifying these connections by terms proportional
to α was never considered before probably because this breaks manifest local N invariance
of the NC connection which depends on mμ only via its curl.
In the presence of torsion the fact that Lσμν is given by (5.5) tells us that we have
no manifest N invariance of the connection. Further, for no value of α can we find such
an invariance. This can be formally solved by adding a new field to the formalism, a
Stu¨ckelberg scalar χ, that cancels the non-invariance. This will be discussed in the next
section. One can also take the point of view as in [30] that we should just accept the fact
that Γ¯ρμν is not N invariant as a mere fact and organize couplings to these geometries and
fields living on it in such a way that the action is N invariant. This is certainly a viable
point of view and agrees with our approach in all these cases where the dependence on χ
can be removed from the theory by field redefinition or simply because it drops out when
one tries to make its appearance explicit.
If one includes χ there is the benefit that one can also deal with theories that do not
have a local U(
1
) symmetry (because there is an explicit dependence on χ so that the
U(
1
) invariance disappears in the Stu¨ckelberg coupling between mμ and χ). This is what
allows us to use fixed TNC background geometries for both Lifshitz field theories (explicit
dependence on χ) as well as Schr¨odinger field theories (no dependence on χ) as discussed
in [23, 24]. The χ field also allows us, as we will see in section 10, to construct two types
of HL actions: those that have a local U(
1
) symmetry without any dependence on χ and
those that have no local U(
1
) because mμ always appears as Mμ = mμ − ∂μχ.
10This statement can be made more precise in the following way. The Hoˇrava-Lifshitz actions of section 9
such as (9.18) take exactly the same form when written in terms of Γ¯ρμν as when expressed in terms of Γˆρμν.
To show this one needs to use the fact that in section 9 it is assumed that τμ is hypersurface orthogonal
which is something that we do not yet impose at this stage. This is because the difference between covariant
derivatives using either one or the other connection involves terms proportional to τμ and since the scalars
in the action are formed by using inverse spatial metrics hμν those terms drop out. The same comments
apply when using the general α of (5.1), i.e. there is no dependence on α.
(5.6)
(5.7)
From now on we will work with (5.6) and simply denote it by Γμν unless specifically
stated otherwise. With this realization of Γρμν the other connections Ωμab and Ωμa are fixed
by the vielbein postulates. For an invariant such as vˆμ the covariant derivatives ∇μ and
Dμ are the same so we can write
∇μvˆν = Dμvˆν = −eνaDμma ,
where we used (2.19) and (2.20) and where Dμma is given by
Dμma = ∂μma
− Ωμabm
b
− Ωμa .
In this section we focussed on making the affine connection G invariant (J invariance
is automatic). It so far is not N invariant. This will be fixed in the next section. We could
have made the connection N but not G invariant by taking Kμν as in (5.5) and Lσμν = 0.
However in this case we are not achieving anything as the connection without Kμν is also
N invariant and so imposing N invariance does not constrain Γρμν . Furthermore since in
the transformation of mμ the G boost parameter λa appears without a derivative, whereas
the N transformation parameter σ appears with a derivative, it is more natural to use mμ
to make various tensors G invariant.
Using the invariants τμ, hμν , vˆμ, hˆμν we can build a non-degenerate symmetric rank
2 tensor with Lorentzian signature gμν that in the case of a relativistic theory we would
refer to as a Lorentzian metric. The metric gμν and its inverse gμν are given by
for which we have
gμν = −τμτν + hˆμν ,
gμν = −vˆμvˆν + hμν ,
gμν vˆμ = τν ,
gμν eaμ = eˆνa .
(5.8)
(5.9)
However the natural Galilean metric structures are τμ and hμν . For example, as we will
see in section 9, gμν does not transform homogeneously under local scale transformations
and so it is not on the same footing as the Riemannian metric in GR.
6
Torsion and the Stu¨ckelberg scalar
In the case of gauging the Poincar´e algebra (appendix A) the torsion is the part of Γμν
that is not fixed by the vielbein postulates. In the case of the Bargmann algebra we see on
the other hand that it is the torsion that is fixed, namely it is given by the antisymmetric
part of (5.1), which reads
It follows that the curvature (2.28) obeys
2Γˆ[ρμν] = −vˆρ (∂μτν − ∂ν τμ) .
Rμν a(P ) = ma (∂μτν − ∂ν τμ) ,
while Rμν (H) = ∂μτν − ∂ν τμ is left arbitrary. Using that Rμν a(P ) transforms as
δ¯Rμν a(P ) = LξRμν a(P ) + λaRμν (H) + λabRμν b(P ) ,
(6.3)
we see that the right hand side of (6.2) transforms in exactly the same way as the left hand
side (ignoring the central extension N ). The right hand side of (6.2) can be matched to
transform correctly under the N transformation by adding the Stu¨ckelberg scalar χ, i.e. by
replacing ma by M a = eμa(mμ − ∂μχ). This explains why in the presence of torsion, i.e.
when ∂μτν − ∂ν τμ 6= 0, we need the scalar χ. In section 10 we will see that there is a similar
field in HL gravity whose couplings are precisely obtained by replacing everywhere mμ by
Mμ = mμ − ∂μχ. From a purely geometrical point of view χ is needed whenever we have
torsion, i.e. when the right hand side of (6.2) is nonzero to ensure correct transformations
under the N generator.
This does not automatically mean that any field theory coupled to such a background
has a nontrivial χ dependence. There are important cases where the χ field can be removed
by a field redefinition or it simply drops out of the action once one tries to make its
appearance explicit. We refer to [23] for field theory examples of the first possibility of
removing χ by field redefinition and to section 10 for a HL action that exhibits the second
property, namely that χ drops out.
The χ field also allows us to make the curvature Rμν (N ) appearing in (4.3), which so
far played no role, visible. This goes via the following commutator
where Dμχ = ∂μχ − mμ and where Rμν (N ) is given by
[Dμ , Dν ]χ = −2Γ[ρμν]Dρχ − Rμν (N ) ,
Rμν (N ) = ∂μmν − ∂ν mμ − 2Ω[μaeν]a .
ρ
terms of Γμν we obtain
We note that by covariance DμDν χ involves the Galilean boost connection Ωμa. Using the
general form of Γρμν given in (3.6) as well as the vielbein postulate (2.16) to express Ωμa in
(6.4)
(6.5)
(6.6)
(6.7)
For the choice Γμν = Γ¯ρμν (5.3), i.e. for Kμν and Lσμν as in (5.5) and (5.5) we find
ρ
Rμν (N ) = ∂μmν − ∂ν mμ − Kμν + vσLσ[μν] .
Rμν (N ) = vσmσ (∂μτν − ∂ν τμ) .
This curvature constraint is in agreement with the curvature constraint (6.2) because it
obeys the transformation rule for the curvatures under Galilean boosts which according
to (2.9) and (2.10) reads δGRμν (N ) = λaRμνa(P ). Again in order that Rμν (N ) remains
inert under N transformations in the presence of torsion we need to replace in Γ¯ρμν (more
precisely in Lσμν as given in (5.5)) mμ by Mμ = mμ − ∂μχ. The field χ is an essential part
of NC geometry with torsion.
The curvature constraints derived here by using the approach of section 2 agree
with [22] where the torsionless case was studied. The analysis of sections 2–6 can thus
be viewed as adding torsion to the gauging of the Bargmann algebra (without adding
dilatations as in [21]). By employing the relation (5.7) between Γ¯ρμν and Γˆρμν we can find
the curvature constraint for Rμν (N ) that relates to this choice of affine connection. The
curvature constraint (6.2) is the same for all affine connections (5.1).
Following [17, 18] we distinguish three cases for the torsion (6.1):
called twistless torsional Newton-Cartan (TTNC) geometry because it is equivalent
to (6.8) which states that the twist tensor is zero.
3. No constraint on τμ which is a novel extension of Newton-Cartan (TNC) geometry.
TTNC geometry goes back to [16] but in that work a conformal rescaling was done to go
to a frame in which there is no torsion. The benefit of adding torsion to the formalism was
first considered in [17, 18] including the case with no constraint on τμ.
We will see below that making NC and TTNC geometries dynamical corresponds to
projectable and non-projectable HL gravity. In this work we will always assume that we
are dealing with TTNC geometry which contains NC geometry as a special case.
For twistless torsional Newton-Cartan (TTNC) geometry we have by definition
hμρhνσ (∂ρτσ − ∂στρ) = 0 .
This implies that the geometry induced on the slices to which τμ is hypersurface orthogonal
is described by (torsion free) Riemannian geometry.
To make contact with the HL literature concerning non-projectable HL gravity it will
prove convenient to define a vector aμ as follows
7
Curvatures
Using that We start by giving some basic properties of the Riemann tensor (2.31) with connection (5.6).
aμ = Lvˆτμ .
In section 8 we will exhibit a coordinate parameterization of aμ (see equations (8.15)
and (8.16)) that will appear more familiar in the context of HL gravity, where this becomes
the acceleration of the unit vector field orthogonal to equal time slices.
For TTNC we have the following useful identities
hμρhνσ (∂ρaσ − ∂σaρ) = hμρhνσ (∇ρaσ − ∇σaρ) = 0 ,
∂μτν − ∂ν τμ = aμτν − aν τμ .
The first of these two identities tells us that the twist tensor (the left hand side) vanishes
which is why we refer to the geometry as twistless torsional NC geometry. The last identity
tells us that aμ describes the TTNC torsion. We will thus refer to it as the torsion vector.
Γρμρ = e−1∂μe ,
(6.8)
(6.9)
(6.10)
(6.11)
(7.1)
where e = det τμ , eaμ , we obtain
Note that because of torsion we have
Rμνρρ = 0 .
Γρρμ = e−1∂μe − vˆρ (∂ρτμ − ∂μτρ) .
From the definition of the Riemann tensor and our choice of connection we can derive the
identity
3R[μνσ]ρ = (∇μvˆρ) (∂ντσ − ∂στν) + (∇σvˆρ) (∂μτν − ∂ντμ)
+ (∇νvˆρ) (∂στμ − ∂μτσ) .
The trace of this equation gives us the antisymmetric part of the Ricci tensor Rμν = Rμρνρ.
The covariant derivative of vˆμ is essentially the extrinsic curvature. Using the
connection (5.6) we find the identity
(δacδbd − δadδbc) R .
where the extrinsic curvature is defined as
∇μvˆρ = −eρaDμma = −hρσKμσ ,
1
Kμν = − 2 Lvˆhˆμν .
For TTNC geometries the antisymmetric part of the Ricci tensor is given by
2Rρ[μν]ρ = (∇ρvˆρ) (aμτν − aντμ) + vˆρ (τν∇μaρ − τμ∇νaρ) ,
using (6.11) and (7.4). We can also derive a TTNC Bianchi identity that reads
3∇[λRμν]σκ = 2Γ[ρμν]Rλρσκ + 2Γ[ρλμ]Rνρσκ + 2Γ[ρνλ]Rμρσκ .
Contracting λ and κ and the remaining indices with vˆμhνσ leads to the identity
0 = e−1∂μ (evˆνhμσRνκσκ) − 12 e−1∂μ (evˆμhνσRνκσκ) + hμρhνσKρσRμκν
κ
where we used (7.3) and (7.5). Since we will mostly work in 2+1 dimensions we focus on
what happens in that case. Using (2.32) we find
e−1∂μ (evˆνhμσRνκσκ) + 1 e−1∂μ (evˆμR) = 0 ,
2
where we used that in 2 spatial dimensions
(7.2)
(7.3)
(7.4)
(7.5)
(7.6)
(7.7)
(7.8)
(7.9)
(7.10)
(7.11)
Coordinate (ADM) parametrizations
Even though we treat the NC fields τμ and hˆμν as independent we can parametrize them
in such a way that gμν in (5.10) is written in an ADM decomposition. Writing
leads to
ds2 = gμν dxμdxν = −N 2dt2 + γij dxi + N idt
dxj + N j dt ,
For the inverse metric (5.11) the ADM decomposition reads
From this we conclude that
The choice (6.8) implies that τμ is hypersurface orthogonal, i.e.
If we fix our choice of coordinates such that τ = t we obtain
Using that τμhμν = 0 and (8.12) we obtain htt = hti = 0 as well as hˆti = γij N j and
ˆ μ
hij = γij . Further using that hμρhˆνρ = δν +vˆμτν we find hij = γij . This in turn tells us that
vˆi = N iN −1, so that htt = hti = 0 leads to vˆt = −N −1. Since vˆμτμ = −1 we also obtain
τt = ψ = N so that hˆtt = γij N iN j . Since htt = hti = 0 we also have vˆt = vt = −N −1
which in turn tells us that hti = htt = 0, so that we find
Furthermore we have hij = γij and vi = 0. For the time component of mμ we obtain
(8.1)
(8.2)
(8.3)
(8.4)
(8.5)
(8.6)
(8.7)
(8.8)
(8.9)
(8.10)
(8.11)
(8.12)
(8.13)
(8.14)
mi = −γij N
.
N j
1
mt = − 2N γij N iN j + N Φ˜ ,
where we used (4.11) or alternatively (4.9) and (4.6). In general τt = N = N (t, x) so that
we are dealing with non-projectable HL gravity. Projectable HL gravity corresponds to
N = N (t) which is precisely what we get when we impose ∂μτν − ∂ν τμ = 0.
In these coordinates the torsion vector (6.9) reduces to
at = N iai ,
ai = N −1∂iN ,
an object such as ∇μ(hμν Xν ) a γ-covariant spatial divergence.
which contains no time derivatives. The determinant e in this parametrization is given by
N √γ where γ is the determinant of γij so that using (7.3) we find Γρρi = ∂i log √γ making
The number of components in gμν in d + 1 space-time dimensions is (d + 1)(d + 2)/2
whereas the total number of components in τμ and hˆμν is (d + 1)(d + 2)/2 + d + 1 − 1 where
the extra d + 1 originate from τμ and the −1 comes from the fact that hˆμν = δabeˆaμeˆbν so that
it has zero determinant. If we furthermore use the fact that τμ is hypersurface orthogonal,
i.e. τμ = ψ∂μτ , we can remove another d − 1 components ending up with (d + 1)(d + 2)/2 + 1
which is one component more than we have in gμν . If we next restrict to coordinate systems
for which τ = t we obtain the same number of components in the ADM decomposition as
we have for our TTNC geometry without Φ˜. Later we will see what the scalars Φ˜ and the
Stu¨ckelberg scalar χ (mentioned below (4.7)) correspond to in the context of HL gravity.
This counting exercise also shows that in general for arbitrary τμ TNC gravity is much
more general than HL gravity. We leave the study of this more general case for future
research. Here we restrict to a hypersurface orthogonal τμ.
We thus see that the field τμ describes many properties that we are familiar with from
the HL literature. For example the TTNC form of τμ in (8.11) agrees with the Khronon
field of [32]. More precisely the Khronon field ϕ of [32] corresponds to what we call τ
and what is called uμ in [32] corresponds to what we call τμ. Further the torsion field ai
that we defined via (6.9) and that has the parametrization (8.16) agrees with the same
field appearing in [32] where it is referred to as the acceleration vector. We will now show
that the generic action describing dynamical TTNC geometries agrees on the nose with
the most general HL actions appearing in the literature.
(8.15)
(8.16)
Hoˇrava-Lifshitz actions
We will consider the dynamics of geometries described by τμ, eaμ and ma (in the next
section we will add Φ˜ and χ) by ensuring manifest G and J invariance and by
constructing in a systematic manner (essentially a derivative expansion) an action for these fields.
Since we demand manifest G and J invariance the generic theory will be described by the
independent fields τμ and hˆμν and derivatives thereof.
For simplicity we will work with twistless torsion and in 2 spatial dimensions with
1 < z ≤ 2. It is straightforward to consider higher dimensions. We will do this in section 12
where we treat the conformal case. A convenient way to organize the terms in the action
is according to their dilatation weight. The dilatation weights of the invariants are given
The second term however brings nothing new because of the identity
hμρaρaν ∇μ (hνσaσ) = − 2
(hμν aμaν )2 − 2
1
1 hμρaμaρ∇ν (hνσaσ) + tot.der. .
(9.9)
Finally we can contract the last three terms in (9.2) with two ∇μ’s leading to the following
set of scalars
∇μvˆμ∇ν vˆν
∇ν vˆμ∇μvˆν
∇ν (hμρaρ) ∇μ (hνσaσ)
There is one other set of scalar terms containing two covariant derivatives that follow by
acting with aμ
where
= hρσ
terms appearing in the list (9.2). This leads to
∇ρ∇σ, which is a dimension 2 operator, on the first two
aμ vˆ
μ
aμ (hμν aν )
2 + z
4 .
Both of these however give nothing new as can be shown by partial integration and upon
using the TTNC identity (6.10).
We are left with the possibility to add scalar curvature terms. To this end we first
introduce a Ricci-type scalar curvature R defined as
which has dilatation weight 2. Using the scalars (9.3) we can thus build the following list
of scalar terms
R = −hμν Rμρν ρ ,
R∇μvˆμ
R
2
R
R∇μ (hμν aν )
Rhμν aμaν
z + 2
2
4
4
4 .
The last term in (9.13) makes it possible to remove ∇μ(hμρaρ)∇ν (hνσaσ) from the list (9.10).
This is due to the identity
∇μ (hμρaρ) ∇ν (hνσaσ) = ∇ν (hμρaρ) ∇μ (hνσaσ) − 2
1
(hμν aμaν )2
where we used (2.30), (2.32), (7.3), (7.11) and partial integrations.
In d = 2 spatial dimensions there are no other curvature invariants other than R. The
reason is that all curvature invariants built out of the tensor Rμνσρ only involve the spatial
Riemann tensor Rμν ab(J ).
The tensor Rabcd = eaμebν Rμν cd(J ) has the same symmetry
properties as the Riemann tensor of a d-dimensional Riemannian geometry. Hence since
(9.10)
HJEP07(215)
(9.11)
(9.12)
(9.13)
hμνaμaν
R
∇μvˆμ∇νvˆν
∇νvˆμ∇μvˆν
hνρaνaρ∇μvˆμ
hνρaρaμ∇νvˆμ
∇μvˆμ∇ν (hνρaρ)
∇νvˆμ∇μ (hνρaρ)
R∇μvˆμ
(hμνaμaν)
R∇μ (hμνaν)
Rhμνaμaν
R
2
2
hμρaμaρ∇ν (hνσaσ)
∇ν (hμρaρ) ∇μ (hνσaσ)
2
2
2z
2z
z + 2
z + 2
z + 2
z + 2
z + 2
4
4
4
4
4
4 .
(9.15)
here d = 2 the only component is the Ricci scalar R. Any other term involving the
curvature tensor contracted with vˆμ or hμνaν can be written as a combination of terms we
already classified using (2.30) and other identities.
We thus conclude that for d = 2 and 1 < z ≤ 2 the scalar terms that can appear in
the action are
Consequently, we arrive at the action S =
Z d3xe hc1hμνaμaν + c2R + c3∇μvˆμ∇νvˆν + c4∇νvˆμ∇μvˆν + c5hνρaνaρ∇μvˆμ
+c6hνρaρaμ∇νvˆμ + c7∇μvˆμ∇ν (hνρaρ) + c8∇νvˆμ∇μ (hνρaρ) + c9R∇μvˆμ
+δz,2 hc10 (hμνaμaν)2 + c11hμρaμaρ∇ν (hνσaσ) + c12∇ν (hμρaρ) ∇μ (hνσaσ)
+ c13R2 + c14R∇μ (hμνaν) + c15Rhμνaμaνii .
(9.16)
The coefficients c1 and c2 have mass dimension z and the coefficients c3 and c4 have mass
dimension 2 − z. All the others are dimensionless. The terms with coefficients c3 and c4
are the kinetic terms because
c3∇μvˆμ∇νvˆν + c4∇νvˆμ∇μvˆν = C hμρhνσKμνKρσ − λ (hμνKμν)2 .
(9.17)
The terms with coefficients c1, c2 and c10 to c15 only involve spatial derivatives and belong
to the potential term V
. They agree with the potential terms in [32, 36, 37, 39] taking into
consideration that we are in 2+1 dimensions. The terms with coefficients c5 to c9 involve
mixed time and space derivatives and are in particular odd under time reversal. Hence in
order to not to break time reversal invariance we will set these coefficients equal to zero.
All other terms are time reversal and parity preserving. We thus obtain
S =
Z d3xe hC hμρhνσKμνKρσ − λ (hμνKμν)
2
− V ,
i
(9.18)
where the potential V is given by
−V = 2Λ + c1hμν aμaν + c2R + δz,2 hc10 (hμν aμaν )2 + c11hμρaμaρ∇ν (hνσaσ)
+ c12∇ν (hμρaρ) ∇μ (hνσaσ) + c13R
2 + c14R∇μ (hμν aν ) + c15Rhμν aμaν i , (9.19)
which also includes a cosmological constant Λ. The kinetic terms in (9.18) display the λ
parameter of [1, 2]. The potential is exactly the same as the 3D version of the potential
given in [32, 36, 37, 39]. We will not impose that V obeys the detailed balance condition.
In the ADM parametrization of section 8 the extrinsic curvature terms in (9.17) are just
γikγjlKij Kkl − λ γij Kij
2
,
where Kij is given by
where Ni = γij N j and ∇i
respect to γij .
1
2N
Kij =
(∂tγij − LN γij ) =
∂tγij − ∇i(γ)Nj − ∇(jγ)Ni ,
1
2N
(γ) is the covariant derivative that is metric compatible with
10
Local Bargmann invariance of the HL action: local U(1) vs. Stu¨ckelberg coupling
The action (9.18) is by construction invariant under local Galilean transformations because
it depends only on the invariants τμ and hˆμν . So far we did not consider the possibility of
adding Φ˜ . The action (9.18) is not invariant under the central extension of the Galilean
algebra. We will now study what happens when we vary mμ in (9.18) as δmμ = ∂μσ. We
have that the connection (5.6) transforms under the central element N of the Bargmann
algebra as
ρ
δN Γμν =
2
1 hρλ [(aμτν − aν τμ) ∂λσ + aλτν ∂μσ + aλτμ∂ν σ]
+ hρλτμτν [∂λ (vˆκ∂κσ) + 2aλvˆκ∂κσ] .
Using that Ωμab is given via (2.13) and (2.16) by
(9.20)
(9.21)
HJEP07(215)
(10.1)
(10.2)
(10.3)
(10.4)
we obtain
This implies that
Ωμab = ebν ∂μeν − Γρμν eρa ,
a
δN Ωμab =
1
2 τμebν eλa (aν ∂λσ − aλ∂ν σ) .
δN Rabcd(J ) = 0 .
Further hμν aν is gauge invariant. Using the above results it can be shown that the whole
potential V in (9.19) is gauge invariant. What is left is to transform the kinetic terms
under N . We have
δN (∇ν vˆμ∇μvˆν − ∇μvˆμ∇ν vˆν ) = −Rvˆμ∂μσ + 2 hμλaμ∇ν vˆν − hμν aμ∇ν vˆλ ∂λσ , (10.5)
that Φ˜ transforms as
where we used (7.10). The first term can be cancelled by adding Φ˜ R to the action using
δN Φ˜ = −vˆμ∂μσ = N −1 ∂tσ − N i∂iσ ,
where in the second equality we expressed the results in terms of the ADM parameterization
of section 8.
In [34] the following U(
1
) transformation was introduced
Together with two new fields A and ν transforming as
with ν called the Newtonian prepotential [34]. We see that the α transformation is none
other than the Bargmann extension (the σ transformation here) as follows from the
identification of mi in (8.13). More precisely we have α = −σ. We thus see that the A and ν
fields can be identified with Φ˜ and χ as follows: A = −N Φ˜ and ν = χ. The term R d3xeRΦ˜
is what in [34] is denoted by R d3x√γAR. If we work in the context of projectable HL
gravity for which aμ = 0 the action (9.18) with λ = 1 can be made U(
1
) invariant by writing
S =
Z
d3xe hC hμρhνσKμν Kρσ − (hμν Kμν )2 − Φ˜ R
− V .
i
(10.10)
However if we work with the non-projectable version or with λ 6= 1 we still need to add
additional terms to make the theory U(
1
) invariant. To see this we use the Stu¨ckelberg
scalar χ that we already mentioned under (4.7) (see also [23]). Using the field χ that
transforms as δχ = σ we can construct the following gauge invariant action (the invariance
is up to a total derivative) for λ = 1
S =
Z
d3xe C (hμρhνσ
− hμν hρσ) Kμν Kρσ − 2aμ∂ν χKρσ + aρ∂σχ∇μ∂ν χ
1
+ 2 aμaρ∂ν χ∂σχ
− CΦ˜ R − V .
(10.6)
(10.7)
(10.8)
(10.9)
(10.11)
The χ dependent terms agree with the result of [36, 37] (eq. (3.8) of that paper).11 We
thus see that when there is torsion aμ 6= 0 we need to introduce a Stu¨ckelberg scalar χ
to make the action U(
1
) invariant. While when there is no torsion we can use (10.10).
11To ease comparison it is useful to note that in the notation of [36, 37] one has the identity
3
1 Gˆijkl 4 (∇i∇jϕ) a(k∇l)ϕ + 2 ∇(iϕ aj)(k∇l)ϕ + 5a(i ∇j)ϕ a(k∇l)ϕ =
1
Gˆijkl (∇i∇jϕ) ak∇lϕ + 2 ajak∇iϕ∇lϕ + tot.der. ,
where in the notation of [36, 37] the field ϕ is what we call χ. We also note that the coefficients of these
terms are dimension independent.
This nicely agrees with the comments made below (6.2). In [34] the χ field is denoted
by ν. This means that we have the following invariance δN mμ = ∂μσ and δN χ = σ. As a
consequence we may simply replace everywhere mμ by Mμ = mμ − ∂μχ. This is consistent
with the observations made in [35] (see in particular eq. (20) of said paper). Essentially
adding the χ field to the action means that we have trivialized the U(
1
) symmetry by
Stu¨ckelberging it or in other words we have removed the U(
1
) transformations all together
(see the next section).
Let us define Kμχν as (7.6) with mμ replaced by Mμ. It can be shown that
hμρhνσKμχν = hμρhνσ
Kρσ − ∇ρ∂σχ − 2 aρ∂σχ − 2 aσ∂ρχ ,
1
1
(10.12)
which is now by construction manifestly U(
1
) invariant. Similarly we can write a manifestly
U(
1
) invariant Φ˜ as
Φχ = Φ˜ + vˆμ∂μχ +
˜
2
1 hμν ∂μχ∂ν χ ,
obtained by replacing mμ by Mμ in Φ˜. Instead of (10.11) we then write
S =
Z
d3xe hC hμρhνσKμχν Kρχσ − hμν Kμχν 2
− ΦχR
− V .
i
It can be checked that this is up to total derivative terms the same as (10.11).
It is now straightforward to generalize this to arbitrary λ and to add for example the
ΩΦ˜ coupling (with Ω being the cosmological constant) considered in [34] leading to
(10.13)
(10.14)
S =
Z
d3xe hC hμρhνσKμχν Kρχσ − λ hμν Kμχν 2
− Φ˜ χ (R − 2Ω) − Vi .
(10.15)
If we isolate the part of the action that depends on χ we find precisely the same answer as
in eq. (3.12) of [37] specialized to 2+1 dimensions.12 We note that for uniformity we have
chosen the coefficient of the Φ˜χR term in (10.14) and (10.15) such that the action has a
U(
1
) symmetry in the absence of torsion, i.e. when the action is independent of χ. In the
presence of χ one can in fact allow for an arbitrary coefficient in front of the Φ˜ χR term.
As a final confirmation that TNC geometry is a natural framework for HL gravity we
will show in section 12 that the conformal HL gravity theories can be obtained by adding
dilatations to the Bargmann algebra, i.e. by considering the Schr¨odinger algebra.
11
A constraint equation
What we have learned is that unless the χ field drops out of the action, as in (10.10) for
the case of projectable HL gravity with λ = 1, we no longer have a non-trivial local U(
1
)
invariance. This is because we can express everything in terms of Mμ which is inert under
the U(
1
). Essentially the fact that we had to introduce a Stu¨ckelberg scalar tells us that
the U(
1
) was not there in the first place.
12This simply means that we can take in the notation of [37] Gij = 0.
because the dilatation weight of vˆμ∂μ Φ˜χ
2
There are several statements in the literature expressing that one can remove a scalar
degree of freedom from the theory by employing the U(
1
) invariance, but since we have just
established that unless we are dealing with (10.10) there is no U(
1
) these statements are
not clear to us. What we will show is that there is a different mechanism that essentially
accomplishes the same effect, via a constraint equation obtained by varying Φ˜ χ in (10.15),
to the claims made in the literature.
Since Φ˜ χ is a field like any other we should, in order to be fully general, allow for
arbitrary couplings to Φ˜χ that do not lead to terms of dimension higher than z + 2. Put
another way the most general HL action can be obtained by writing down the most general
action depending on τμ, hˆμν and Φ˜ χ containing terms up to order (dilatation weight) z + 2.
The first thing to notice is that we typically cannot write down a kinetic term for Φ˜ χ
is 6z − 4 which is larger than z + 2 whenever
term of the form Φ˜ 2χ.
z > 6/5. The same is true for Kvˆμ∂μΦ˜ χ while a term like vˆμ∂μ Φ˜χ or what is the same
upon partial integration KΦ˜ χ breaks time reversal invariance. Let us assume that we have
a z value larger than 6/5 so that we cannot write a kinetic term. This means that Φ˜ χ will
appear as a non-propagating scalar field.
Let us enumerate the possible allowed couplings to Φ˜ χ. Starting with the kinetic
terms we can have schematically Φ˜χαK2 where by K2 we mean both ways of contracting
the product of two extrinsic curvatures. In order for this term to have a dimension less
than or equal to z + 2 we need that α ≤ 2(2z−−z1) . It follows that for z > 4/3 we need α < 1.
Consider next a term of the form Φ˜ βχX where X is any term of dimension 2. The condition
that the weight does not exceed z + 2 gives us β ≤ 2(zz−1) which means that if z > 4/3
we need β < 2. Finally we can have terms of the form Φ˜γχ where γ ≤ 2(z−1) so that for
2+z
z > 8/5 we need that γ < 3. In particular it is allowed for all values of 1 < z ≤ 2 to add a
Since for z > 6/5 we are not allowed to add a kinetic term for Φ˜ χ we can integrate
it out. We demand that the resulting action after integrating out Φ˜ χ is local. This puts
constraints on what α, β and γ can be since they influence the solution for Φ˜ χ. We assume
here that α, β and γ are non-negative integers. We will be interested in values of z close to
z = 2 so we assume that z > 8/5. In that case we have the following allowed non-negative
integer values: α = 0, β = 0, 1 and γ = 0, 1, 2. In other words we can add the following Φ˜ χ
dependent terms
Φχ d1 + d2R + d3∇μ (hμν aν ) + d4hμν aμaν + d5 Φ˜χ .
˜
(11.1)
There are now two cases of interest: either d5 6= 0 or d5 = 0. When d5 6= 0 we can solve
for Φ˜χ and substitute the result back into the action. The resulting action will be of the
same form as (9.18) where all the terms originating from solving for Φ˜χ and substituting
the result back into the action can be absorbed into the potential terms by renaming the
that case the equation of motion of Φ˜ χ leads to the constraint equation
coefficients in V
. The other possibility that d5 = 0 leads to a rather different situation. In
d1 + d2R + d3∇μ (hμν aν ) + d4hμν aμaν = 0 .
(11.2)
HJEP07(215)
The remaining equations of motion for τμ etc. will depend on Φ˜ χ because there is no local
symmetry (in particular no U(
1
)) that allows us to gauge fix this field to zero. Since there
is no kinetic term for Φ˜χ, and hence its value will not be determined dynamically, we fix
it by adding a coupling to an auxiliary field. Recall that for any value of z in the range
1 < z ≤ 2 it is allowed by the effective action approach to add a term proportional to Φ˜ 2χ.
Consider now the following action S =
Z
d3xe hΦ˜ χ indep. part+Φ˜ χ (d1 + d2R + d3∇μ (hμν aν )+d4hμν aμaν )+λΦ˜ 2χi , (11.3)
where crucially now λ is an auxiliary field, which has the property that its equation of
motion tells us that Φ˜ χ = 0 and further the equation of motion of Φ˜ χ will lead to the
constraint equation (11.2), which is a more general version of the constraint equation used
in [34] and related works. Since Φ˜ χ = 0 the Φ˜ χ dependent terms do not affect the remaining
equations of motion. This essentially accomplishes that Φ˜χ is not present in the theory
and that we have the constraint equation (11.2). More generally we should think of Φ˜ χ as
a background field whose value can be set to be equal to some fixed function f . This is
accomplished by writing instead of (11.3) the following
S =
Z
d3xe Φ˜χ indep. part +
˜
Φχ − f (d1 + d2R + d3∇μ (hμν aν ) + d4hμν aμaν )
+λ Φ˜χ − f
2
.
(11.4)
The λ equation of motion enforces the background value Φ˜ χ = f , the equation of motion of
Φ˜χ leads again to (11.2) while the remaining equations of motion involve terms depending
on f through the variation of terms linear in f .
12
Conformal HL gravity from the Schr¨odinger algebra
In this section we will work with an arbitrary number of spatial dimensions d. In order
to study conformal HL actions we add dilatations to the Bargmann algebra of section 4
and study the various conformal invariants that one can build. To this end we use the
for now the special conformal transformations that will be introduced later)13
connection Aμ that takes values in the Schr¨odinger algebra (where for z = 2 we leave out
Aμ = Hτμ + Paeaμ + Gaωμa +
1
2 Jabωμab + N m˜ μ + Dbμ ,
(12.1)
where the new connection bμ is called the dilatation connection. The reason that we
renamed the connections in (12.1) as compared to (4.1) is because the dilatation generator D
is not central so that it modifies the transformations under local D transformations as
compared to how say Ωμa and ωμab would transform using (2.13), (2.16) and (5.1). The
transfor13Compared to e.g. [19, 21] we have interchanged the field mμ appearing in front of N in the Bargmann
algebra and the field m˜μ appearing in front of N in the Schr¨odinger algebra, see also footnote 9.
mation properties and curvatures of the various fields follow from the Schr¨odinger algebra:
[D , H] = −zH ,
[D , Ga] = (z − 1)Ga ,
[Jab , Jcd] = δacJbd − δadJbc − δbcJad + δbdJac .
where we write (without loss of generality)
with now
and we define δ¯Aμ as
where Fμν is the curvature
We perform the same steps as before (see (2.3) and onwards), namely we consider the
adjoint transformation of Aμ, i.e.
2 Jabλab + N σ + DΛD ,
δ¯Aμ = δAμ − ξν Fμν = LξAμ + ∂μΣ + [Aμ , Σ] ,
(12.2)
(12.3)
(12.4)
(12.5)
(12.6)
(12.7)
(12.8)
Fμν = ∂μAν − ∂ν Aμ + [Aμ , Aν ]
= HR˜μν (H) + PaR˜μν a(P ) + GaR˜μν a(G) +
+ DR˜μν (D) ,
1
2 JabR˜μν ab(J ) + N R˜μν (N )
where we put tildes on the curvatures to distinguish them from those given in sections 2
and 4. From this we obtain among others that the dilatation connection bμ transforms as
δ¯bμ = Lξbμ + ∂μΛD .
The following discussion closely follows section 4 of [21]. We will use this bμ connection
to rewrite the covariant derivatives (2.12) and (2.13) in a manifestly dilatation covariant
manner.
As a note on our notation we remark that, now that we have learned that we should
work with Mμ = mμ − ∂μχ we take it for granted that we have replaced everywhere mμ
by Mμ and we from now on suppress χ labels as in (10.12) and (10.13). The Schr¨odinger
algebra for general z tells us that the dilatation weights of the fields are as in table 1 while
mμ and χ (and thus Mμ) have dilatation weight z − 2. This also agrees with the weights
assigned to A and ν in [34].
Coming back to the introduction of bμ, to make expressions dilatation covariant we take
Γ¯ρμν of equation (5.3) and replace ordinary derivatives by dilatation covariant ones leading
Dμτν = ∂μτν − Γ˜ρμντρ − zbμτν = 0 ,
Dμeνa = ∂μeνa − Γ˜ρμνeρa − ωμaτν − ωμabeνb − bμeνa = 0 .
The ωμa and ωμab connections are such that they can be written in terms of Ωμa and Ωμab
together with bμ dependent terms such that all the bμ terms drop out on the right hand side
of (12.10) and (12.11) when expressing it in terms of the connections Γρμν, Ωμa and Ωμab.
The field Mμ = mμ − ∂μχ can be expressed in terms of the Schr¨odinger connection m˜μ
as follows. According to (12.2) and (12.6) the Schr¨odinger connection m˜ μ transforms as
δ¯ m˜μ = Lξ m˜μ + ∂μσ + λaeμa + (z − 2) (σbμ − ΛD m˜μ) .
to a new connection Γ˜ρμν that is invariant under the Ga, Jab, N and D transformations and
which is given by [21]
Γ˜ρμν = −vˆρ (∂μ −zbμ) τν + 1 hρσ (∂μ −2bμ) h¯νσ +(∂ν −2bν) h¯μσ −(∂σ − 2bσ) h¯μν . (12.9)
For the most part of this section we will work with Γ¯ρμν and its dilatation covariant
generalization Γ˜ρμν. The final scalars out of which we will build the HL action, i.e. for dynamical
TTNC geometries, are such that it does not matter whether we use Γ¯ρμν or Γˆρμν which are
related via (5.7).
and (2.13) as follows
With the help of bμ and Γ˜ρμν we can now rewrite the covariant derivatives (2.12)
The Stu¨ckelberg scalar χ transforms as
A Schr¨odinger covariant derivative Dμχ is given by
δ¯χ = Lξχ + σ + (2 − z)ΛDχ .
Dμχ = ∂μχ − m˜μ − (2 − z)bμχ .
Defining Mμ = −Dμχ = mμ − ∂μχ we see that Mμ transforms as
and that
Hence the dilatation covariant derivative of Mμ reads
δ¯Mμ = LξMμ + eμaλa + (2 − z)ΛDMμ ,
mμ = m˜μ + (2 − z)bμχ .
DμMν = ∂μMν − Γ˜ρμνMρ − (2 − z)bμMν − ωμaeνa .
(12.10)
(12.11)
(12.12)
(12.13)
(12.14)
(12.15)
(12.16)
(12.17)
(12.18)
The torsion Γ˜[ρμν] has to be a G, J , N and D invariant tensor. With our TTNC field
content the only option is to take it zero, i.e. Γ˜ρμν becomes torsionless [21]. This means
that different from the relativistic case the bμ connection is not entirely independent, but
instead reads
bμ = 1 vˆρ (∂ρτμ − ∂μτρ) − vˆρbρτμ = z aμ − vˆρbρτμ .
1
z
Let Xρ be a tensor with dilatation weight w, i.e.
A dilatation covariant derivative is given by
δDXρ = −wΛDXρ .
∇˜ νXρ + wbνXρ ,
where ∇˜ ν is covariant with respect to Γ˜νρμ as given in (12.9). Let us compute the
commutator
where
∇˜ μ + wbμ
∇˜ ν + wbν Xρ − (μ ↔ ν)
= −R˜μνλρXλ + w (∂μbν − ∂νbμ) Xρ ,
R˜μνλρ = −∂μΓ˜νρλ + ∂νΓ˜ρμλ − Γ˜ρμσ Γ˜νσλ + Γ˜νρσΓ˜σμλ .
The introduction of the bμ connection has led to a new component vˆμbμ as visible
in (12.18). We can introduce a special conformal transformation (denoted by K) that allows
us to remove this component. Hence we assign a new transformation rule to bμ namely
δK bμ = ΛK τμ .
Under special conformal transformations we have
δK Γ˜ρμν = ΛK (z − 2)vˆρτμτν − δμτν − δν τμ .
ρ ρ
In order that ∇˜ μ + wbμ
∇˜ ν + wbν Xρ transforms covariantly we define the K-covariant
derivative
D˜μ + wbμ
∇˜ ν + wbν Xρ =
∇˜ μ + wbμ
∇˜ ν + wbν Xρ
ρ ρ
− wfμτνXρ − fμ (z − 2)vˆρτντλ − δν τλ − δλτν Xλ ,
where fμ is a connection for local K transformations that transforms as [21]
δ¯fμ = Lξfμ + ∂μΛK − zΛDfμ + zΛK bμ .
In order not to introduce yet another independent field fμ (recall that we are trying
to remove vˆμbμ) we demand that fμ is a completely dependent connection that transforms
as in (12.26). This is in part realized by setting the curvature of the dilatation connection
bμ equal to zero, i.e. by imposing
Rˇμν(D) = ∂μbν − ∂νbμ − fμτν + fντμ = 0 .
This fixes all but the vˆμfμ component of fμ. This latter component will be fixed later by
equation (12.42). The notation is such that a tilde refers to a curvature of the δ¯
transformation (12.6) without the K transformation while a curvature with a check sign refers to
(12.19)
(12.20)
(12.21)
(12.22)
(12.23)
(12.24)
(12.25)
(12.26)
(12.27)
HJEP07(215)
a curvature that is covariant under the δ¯ transformations with the K transformation. We
note that while for the Schr¨odinger algebra, i.e. with the δ transformations (12.3) we can
only add special conformal transformations when z = 2 while for the (different) group of
transformations transforming under δ¯ we can define K transformations for any z [21].
Taking the commutator of (12.25) we find
˜
∇ν + wbν Xρ
− (μ ↔ ν) = −RˇμνλρXλ ,
where Rˇμνλρ is given by
Rˇμνλρ = R˜μνλρ + (z − 2)vˆρτλ (fμτν − fν τμ) − δν τλfμ + δμρτλfν
ρ
HJEP07(215)
− δλρ (fμτν − fν τμ) .
Under K transformations the curvature tensor Rˇμνλρ transforms as
δK Rˇμνλρ = ΛK [−(z − 2)τλτν Dμvˆρ + (z − 2)τλτμDν vˆρ] .
Besides this property, the tensor Rˇμνλρ is by construction invariant under D, G, N and J
transformations.
Using the vielbein postulates (12.10) and (12.11) we can write
Γ˜ρμν = −vρ (∂μτν − zbμτν ) + eρa ∂μeν − ωμaτν − ωμabebν − bμeνa .
a
With this result we can derive
where R˜μνcd(J ) and Rˇμνc(G) are given by
Rˇμνσρ = −eρdecσR˜μνcd(J ) + eρcτσRˇμνc(G) ,
R˜μν ab(J ) = 2∂[μων]ab
− 2ω[μcaων]bc ,
Rˇμν a(G) = R˜μν a(G) − 2f[μ eν]a + (z − 2)τν]M a
= 2∂[μων]a − 2ω[μabων]b − 2(1 − z)b[μων]a
− 2f[μ eν]a + (z − 2)τν]M a .
We next present some basic properties of Rˇμνσρ. The first is
Rˇμνρρ = 0 ,
Rˇ[μνσ]ρ = 0 .
R˜[μν ab(J )eρ]b + Rˇ[μν a(G)τρ] = 0 .
Rˇcaa(G) + vμR˜μaac(J ) = 0 ,
Equations (12.32) and (12.36) together give us the Bianchi identity
By contracting this with vμeν ceρa we find
(12.28)
(12.29)
(12.30)
(12.31)
(12.32)
(12.33)
(12.34)
(12.35)
(12.36)
(12.37)
(12.38)
= −vˆμ (Rμaa(G) + 2M cRμaac(J )) − M b (Rbaa(G) + M cRbaac(J )) .
(12.41)
We now turn to the question what vˆσvˆν Rˇσν should be equal to. Following [21] we will take
this to be equal to
1
2d
vˆσvˆν Rˇσν =
(z − 2) (hμν
DμMν )2 ,
because the right hand side has the exact same transformation properties under all local
symmetries as vˆσvˆν Rˇσν . The combination of Rˇμν (D) = 0 together with (12.42) fixes fμ
entirely in terms of τμ, eaμ, mμ and χ in such a way that it transforms as in (12.26).
Using that
where Kμν is the extrinsic curvature, we see that
hμρhνσ
D(μMν) = hμρhνσ
Kμν + vˆλbλhˆμν ,
hμρhνσ
D(μMν)D(ρMσ) − d
1
(hμν
DμMν )2 ,
is invariant under the K transformation because the term vˆμbμ cancels out from the above
difference. Another scalar quantity of interest is
hμν Rˇμν = −R˜abab(J ) ,
which is K invariant and has dilatation weight 2. With these ingredients we can build a
z = d conformally invariant Lagrangian
and by contracting (12.37) with eμbeν aeρc we obtain
The two identities (12.36) and (12.35) imply that
R˜baac(J ) − R˜caab(J ) = 0 .
Rˇρ[νσ]ρ = 0 .
We define Rˇνσ = Rˇνρσρ.
Using the identity (12.38) we can derive
vˆσvˆν Rˇσν = −vˆν (Rνaa(G) + M cRνaac)
ωμab = Ωˆμab + eνbbν eˆμ − eνabν eˆbμ ,
a
L = e A
hμρhνσKμν Kρσ − d
1
(hμν Kμν )
2
+ B hμν Rˇμν
d .
(12.46)
This is an example of a Lagrangian for non-projectable HL gravity that is conformally
invariant.
and Γ¯ρμν given in (5.3) which reads
The quantity hμν Rˇμν can be expressed in terms of R and the torsion vector aμ defined
in sections 6 and 7 as follows. Solving (12.11) for ωμab and using the relation between Γ˜ρμν
Γ˜ρμν = Γ¯ρμν + zvˆρbμτν − hρσ bμh¯νσ + bν h¯μσ − bσh¯μν ,
we obtain (12.11), via
(12.39)
(12.40)
(12.42)
(12.43)
(12.44)
(12.45)
(12.47)
(12.48)
where we used that ωμab and Ωμab are related, as follows from the vielbein postulates (2.13),
(2.16) and where we furthermore used that for TTNC Ω¯ μab = Ωˆ μab as follows from (5.7)
and the TTNC relation (6.10). In the relation Ω¯μab = Ωˆ μab the connection Ω¯μab is found by
employing the vielbein postulate expressed in terms of Γ¯ρμν and likewise Ωˆ μab is obtained by
using the vielbein postulate written in terms of Γˆρμν . Then using (12.33) and (12.48) we find
hμν Rˇμν = −R˜cdcd(J ) = −R + 2(d − 1)∇μ (hμν aν ) − (d − 1)(d − 2)hμν aμaν ,
(12.49)
where we used (12.18) and Rcdcd(J ) = R which is merely a definition of R.
By fully employing the local Schr¨odinger algebra we arrive at the conformally invariant
dd+1xe A
hμρhνσKμν Kρσ − d
1
+ B (R − 2(d − 1)∇μ (hμν aν ) + (d − 1)(d − 2)hμν aμaν )d .
For z = d the dilatation weight of Φ˜ is given by 2(d − 1) so that the terms
− aΦ˜ (R − 2(d − 1)∇μ (hμν aν ) + (d − 1)(d − 2)hμν aμaν ) + bΦ˜ d−1 ,
d
can be added to the action in a conformally invariant manner. Assuming b 6= 0 we can
integrate out Φ˜ which leads to the action (12.50) with a different constant B. The case
with b = 0 can be viewed as a constrained system as discussed in section 11. The
integrand of (12.50) has been obtained in Lifshitz holography and field theory using different
techniques and found to describe the Lifshitz scale anomaly [4, 18, 49–51] where A and
B play the role of two central charges. In [18] it was shown that for d = z = 2 the
integrand of (12.50) together with (12.51) for specific values of a and b arises from the
(Scherk-Schwarz) null reduction of the AdS5 conformal anomaly of gravity coupled to an
(12.50)
(12.51)
axion.
13
Discussion
We have shown that the dynamics of TTNC geometries, for which there is a hypersurface
orthogonal foliation of constant time hypersurfaces, is precisely given by non-projectable
Hoˇrava-Lifshitz gravity. The projectable case corresponds to dynamical NC geometries
without torsion. One can build a precise dictionary, between properties of TNC and HL
gravities, which we give below in table 2.
We conclude with some general comments about interesting future research directions.
TNC geometries have appeared so far as fixed background geometries for
nonrelativistic field theories and hydrodynamics [23–26, 30, 52–54] as well as in holographic
setups based on Lifshitz bulk space-times [17–19, 21, 23]. In all these cases the TNC
geometry is treated as non-dynamical. This is a valid perspective provided the backreaction
onto the geometry can be considered small, e.g. a small amount of energy or mass density
should not lead to pathological behavior of the geometry when allowing it to backreact.
TNC gravity
twistless torsion: hμρhνσ (∂μτν − ∂ν τμ) = 0
no torsion: ∂μτν − ∂ν τμ = 0
τμ = ψ∂μτ
τ invariant under Galilean
tangent space group
torsion vector aμ
TNC invariant: −τμτν + hˆμν
τi = 0
τt
mi = −N −1Ni
ˆ
hij
on mμ and χ
∇μvˆν
scalar Φ˜ in mt = − 21N γij N iN j + N Φ˜
Stu¨ckelberg scalar χ
Bargmann central extension acting
two scalar invariants ∇μvˆμ∇ν vˆν and ∇ν vˆμ∇μvˆν
allowed by local Galilean symmetries
Effective action organized by
Schr¨odinger representations
Local Schr¨odinger invariance
general torsion: no constraint on τμ
metric with Lorentz signature gμν
HL gravity
non-projectable
projectable
scalar khronon ϕ in uμ [32]
foliation breaks local Lorentz
invariance
acceleration aμ [32]
ADM decomposition lapse N
ADM shift vector Ni
metric on constant t slices γij
N −1A [34]
Newtonian prepotential ν [34]
local U(
1
) acting on A, Ni and ν
extrinsic curvature
the λ parameter in the kinetic term
Dimensions: [N ] = L0, [γij ] = L0,
[N i] = L1−z, [A] = L−2(z−1)
conformal HL actions (invariant
under anisotropic Weyl rescalings)
vector khronon [3]
This renders the question of the consistency of HL gravity in the limit of small fluctuations
around flat space-time of crucial importance for applications of TNC geometry to the realm
of non-relativistic physics.
In this light we wish to point out that (in the absence of a cosmological constant) the
ground state is not Minkowski space-time but flat NC space-time which has different
symmetries than Minkowski space-time as worked out in detail in [23]. It would be interesting
to work out the properties of perturbations of TTNC gravity around flat NC space-time. In
particular we have shown that generically there is no local U(
1
) symmetry in the problem
but that rather one can either integrate out Φ˜ χ without modifying the effective action in
an essential way or in such a way that it imposes a non-trivial constraint on the spatial
part of the geometry. It would also be interesting to study the theory from a Hamiltonian
perspective and derive the first and second class constraints and compare the resulting
counting of degrees of freedom with the linearized analysis.
Since it is well understood how to couple matter to TNC geometries the question of how
to couple matter to HL gravity can be readily addressed in this framework. For example it
would be interesting to find Bianchi identities for the TTNC curvature tensor (as studied
in section 7) in such a way that they are compatible with the on-shell diffeomorphism Ward
identity for the energy-momentum tensor as defined in [19, 23, 24]. We emphasize once
more that matter systems coupled to TNC geometries can have but do not necessarily
need to have a particle number symmetry [19, 23]. It would be important to study what
(13.1)
(13.2)
(13.3)
the fate of particle number symmetry is once we make the geometry dynamical. In the
matter sector particle number symmetry comes about as a gauge transformation acting
on Mμ in such a manner that the Stu¨ckelberg scalar χ can be removed from the matter
action [19, 23] making this formulation consistent with [30]. We have seen in section 10
that generically the χ field cannot be removed from the actions describing the dynamics
of the TNC geometry. Hence, it seems that the dynamics of the geometry breaks particle
number symmetry except when we use the model (10.10) for projectable HL gravity with
λ = 1 in which case the central extension of the Bargmann algebra is a true local U(
1
)
symmetry and the χ field does not appear in the HL action.
Another interesting extension of this work is to consider the case of unconstrained
torsion, i.e. TNC gravity, in which case τμ is no longer restricted to be hypersurface
orthogonal. In table 2 we refer to this as the vector khronon extension in the last row.
The main difference with TTNC geometry is that now the geometry orthogonal to τμ is
no longer torsion free Riemannian geometry but becomes torsionful. This extra torsion is
described by an object which we call the twist tensor (see e.g. [21]) denoted by Tμν and
Tμν =
δμρ + τμvˆρ (δνσ + τν vˆσ) (∂ρτσ − ∂στρ) .
Therefore apart from the fact that now the τμ appearing in the actions of sections 9–12 is no
longer of the form ψ∂μτ but completely free, we can also add additional terms containing
the twist tensor Tμν . Another such tensor is T(a)μν (see again [21] where it was denoted by
T(b)μν ) which is defined as
T(a)μν =
δμρ + τμvˆρ (δνσ + τν vˆσ) (∂ρaσ − ∂σaρ) .
Hence we can add for example a term such as
Tμν Tρσhμρhνσ ,
which has weight 4 − 2z so that it is relevant for z > 1. In fact for z = 2 this term has
weight zero and so one can add an arbitrary function of the twist tensor squared. In the
IR however the two-derivative term dominates. Another aspect that would be worthwhile examining using our results is whether one could learn more about non-relativistic field theories at finite temperature using holography – 36 –
for HL gravity [3–5, 49, 55, 56]. Independently of whether HL gravity is UV complete,
assuming it makes sense as a classical theory it may be a useful tool to compute properties
such as correlation functions of the (non-relativistic) boundary field theory. In particular,
this implies that there must exist bulk gravity duals to thermal states of the field theory,
i.e. classical solutions of HL gravity that resemble black holes as we know them in general
relativity. In light of this it would be interesting to re-examine the status of black hole
solutions in HL gravity (see e.g. [57–59]). Moreover, it is expected that in a long-wave
length regime some version of the fluid/gravity correspondence should exist, enabling the
computation of for example transport coefficients in finite temperature non-relativistic field
theories on flat (or more generally curved) NC backgrounds.
TNC geometry also appears in the context of WCFTs [60] as the geometry to which
these SL(2) × U(
1
) invariant CFTs couple to. This was called warped geometry and
corresponds to TNC geometry in 1 + 1 dimensions with z = ∞ (or z = 0 if one interchanges
the two coordinates). In that case there is no spatial curvature so the entire dynamics is
governed by torsion. It would be interesting to write down the map to the formulation
in [60] and furthermore explicitly write the HL actions for that case.
It would also be interesting to explore the relation of TNC gravity to Einstein-aether
theory. It was shown in [38] that any solution of Einstein-aether theory with hypersurface
orthogonal τμ is a solution of the IR limit of non-projectable HL gravity. It would thus
be natural to expect that any solution of Einstein-aether theory with unconstrained τμ
is a solution to the IR limit of TNC gravity. In view of the relation [61, 62] between
causal dynamical triangulations (CDT) and HL quantum gravity, both involving a global
time foliation, there may also be useful applications of TNC geometry in the context of
CDT [63]. Finally, since HL gravity is connected to the mathematics of Ricci flow (see
e.g. [64]), examining this from the TNC perspective presented in this paper could lead to
novel insights.
Acknowledgments
We would like to thank Ioannis Bakas, Jan de Boer, Diego Hofman, Kristan Jensen, Cindy
Keeler and Elias Kiritsis for valuable discussions. The work of JH is supported by the
advanced ERC grant ‘Symmetries and Dualities in Gravity and M-theory’ of Marc Henneaux.
The work of NO is supported in part by the Danish National Research Foundation project
“New horizons in particle and condensed matter physics from black holes”. We thank the
Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial
support during the completion of this work.
Gauging Poincar´e
In this appendix we briefly discuss the gauging of the Poincar´e algebra to show the power
of the method in a more familiar context. Consider the Poincar´e algebra whose generators
are Pa and Mab satisfying the commutation relations
[Mab , Pc] = ηacPb − ηbcPa ,
[Mab , Mcd] = ηacMbd − ηadMbc − ηbcMad + ηbdMac .
This connection transforms in the adjoint as
where Λ is given by
where
with σab = ξμωμab + λab. Next we define δ¯Aμ as
Λ = ξμAμ + Σ ,
Σ =
Mabλab ,
δ¯Aμ = δAμ − ξν Fμν = LξAμ + ∂μΣ + [Aμ , Σ] ,
where the second equality is an identity and where Fμν is the curvature
We introduce the Lie algebra valued connection Aμ given by
What we have done so far is to make the Poincar´e transformations local. However
we would like to connect this to a set of transformations that replace local space-time
translations by diffeomorphisms. This can be achieved as follows. We define a new set
of local transformations that we denote by δ¯. The main step is to replace the parameters
in Λ corresponding to local space-time translations, i.e. ζa by a space-time vector ξμ via
ζa = ξμeaμ. This can achieved by the following way of writing Λ
in which we have
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
Aμ = Paeaμ +
Mabωμab .
Λ = Paζa +
Mabσab .
MabRμν ab(M ) ,
Rμν a(P ) = 2∂[μeν] − 2ω[μabeν]b ,
a
Rμν ab(M ) = 2∂[μων]ab
− 2ω[μcaων]bc .
transforms as a vielbein while the connection ωμab associated with the Lorentz boosts Mab
become the spin connection coefficients.
In order to define a covariant derivative on the space-time we first introduce a covariant
derivative Dμ via
Dμeνa = ∂μeν − Γρμν eρa − ωμabebν ,
a
which transforms covariantly under the δ¯ transformations. The affine connection Γρμν
transforms under the δ¯ transformations as
so that it is inert under the local Lorentz (tangent space) transformations. We will now
relate the properties of the curvatures Rμν a(P ) and Rμν ab(M ) to those of Γρμν . This goes
via the vielbein postulate which reads
δ¯Γρμν = ∂μ∂ν ξρ + ξσ∂σΓρμν + Γρσν ∂μξσ + Γρμσ∂ν ξσ − Γσμν ∂σξρ ,
(A.13)
(A.12)
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
ρ
Γ[μν] to the other side we obtain
relating Γρμν to ωμab. Taking the antisymmetric part of the vielbein postulate and moving
Dμeνa = ∂μeν − Γρμν eρa − ωμabebν = 0 ,
a
Rμν a(P ) = 2∂[μeν] − 2ω[μabeν]b = 2Γ[ρμν]eρ .
a a
∇μ (containing only the connection Γρμν ) leading to
From this we conclude that the curvature Rμν a(P ) is the torsion tensor. To identify the
other curvature tensor Rμν ab(M ) we compute the commutator of two covariant derivatives
[∇μ , ∇ν ]Xρ = RμνρσXσ − 2Γ[σμν]∇σXρ ,
∇ρgμν = 0 .
where Rμνρσ is the Riemann curvature tensor
that is related to Rμν ab(M ) (as follows from the vielbein postulate) via
Rμνρσ = −eρaebσRμν ab(M ) ,
so that Rμν ab(M ) is the Riemann curvature 2-form. The vielbein postulate, because of
the fact that ωμab is antisymmetric in a and b, also tells us that the metric gμν = ηabeaμebν ,
which is the unique Lorentz invariant tensor we can build out of the vielbeins, is covariantly
constant, i.e.
As is well known this fixes completely the symmetric part of the connection making it equal
to the Levi-Civit`a connection plus torsion terms which are left unfixed. The common
choice in GR to work with torsion-free connections then implies that from the gauging
perspective one imposes the curvature constraint Rμν a(P ) = 0. This in turn makes ωμab
a fully dependent connection expressible in terms of the vielbeins and their derivatives.
Without fixing the torsion eaμ and ωμab remain independent.
This article is distributed under the terms of the Creative Commons
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