Preprint typeset in JHEP style - HYPER VERSION

Lifshitz Gravity for Lifshitz Holography

Tom Griffina,b , Petr Hořavaa,b and Charles M. Melby-Thompsonc
Berkeley Center for Theoretical Physics and Department of Physics
University of California, Berkeley, CA 94720-7300, USAaPhysics Division, Lawrence Berkeley National Laboratory
Berkeley, CA 94720-8162, USAb

Kavli Institute for the Physics and Mathematics of the Universe
University of Tokyo, Kashiwa 277-8583, Japanc

Abstract: We argue that Hořava-Lifshitz (HL) gravity provides the minimal holographic
dual for Lifshitz-type field theories with anisotropic scaling and dynamical exponent z. First
we show that Lifshitz spacetimes are vacuum solutions of HL gravity, without need for additional matter. Then we perform holographic renormalization of HL gravity, and show how
it reproduces the full structure of the z = 2 anisotropic Weyl anomaly in dual field theories
in 2 + 1 dimensions, while its minimal relativistic gravity counterpart yields only one of two
independent central charges in the anomaly.

The concepts of scaling and the renormalization group have played a central role inorganizing our understanding of quantum field theory (QFT) and statistical systems for halfa century. Here we will focus on systems in D + 1 spacetime dimensions, which exhibit scaling
anisotropic between time and space,t ! bz t,

xi ! bxi ,i = 1, . . . D,

(1)with the degree of anisotropy measured by the dynamical exponent z. Systems with such
Lifshitz scaling appear frequently in quantum and statistical field theory of condensed matter systems [1], especially in the context of Lifshitz multicritical points, and in nonequilibrium statistical mechanics. More recently, in a seemingly unrelated development, anisotropic
Lifshitz-type scaling (1) has played a central role in the new approach to quantum gravity
initiated in [2, 3] and commonly referred to as Hořava-Lifshitz (HL) gravity.
Since AdS/CFT correspondence taught us that many relativistic QFTs have relativistic
gravity duals, it seems natural to expect that the two disparate applications of Lifshitz scaling– nonrelativistic QFT on one hand and HL gravity on the other – should similarly be related
by a holographic duality. The background geometry that captures the spacetime symmetries
of QFTs with Lifshitz scaling (1) is easy to find; it is given by the Lifshitz spacetime [4] in
D + 2 dimensions,✓ ◆2⇣ r ⌘2z⇣ r ⌘2
`
2
2
i
i
ds =
dt +
dx dx +
dr2 .
(2)
`
`
r
(From now on, we will set its radius of curvature ` = 1 for convenience.) The holographic
gravity duals of Lifshitz-type QFTs should therefore have (2) as their solution.
Until now, the overwhelming share of work on Lifshitz holography (starting with [4])does not use HL gravity – it uses relativistic bulk gravity coupled to matter instead. In therelativistic case, the coupling to matter is necessary, as the Lifshitz spacetime with z 6= 1
does not solve the Einstein equations in the vacuum. However, we stress that another natural
option is available: Instead of adding ad hoc matter to Einstein gravity so that (2) becomesa solution, one can modify gravity itself.
Perhaps the most popular relativistic model for Lifshitz holography, proposed in [5],consists of Einstein gravity (described by the bulk metric Gµ⌫ , in coordinates y µ = (t, xi , r),with µ = 0, . . . D + 1), coupled to a massive vector Aµ :
⇢
Z
p
1
1
1 2
D
Srel =
dt d x dr
G R 2⇤
Fµ⌫ F µ⌫m Aµ Aµ + surface terms. (3)
16⇡GN
4
2

The Lifshitz geometry (2) is a solution for an appropriate condensate of A0 and an appropriate
choice of ⇤ and m.
In this paper, we will follow the alternate path, and show that the Lifshitz spacetime is avacuum solution of minimal HL gravity, with no additional matter. The preferred foliation of
the Lifshitz spacetime, required for its embedding into HL gravity, is simply the foliation byleaves of constant t. We will often split the bulk coordinates y µ into time t plus D + 1 spatial–1–

coordinates y a = (xi , r), a = 1, . . . D, and write the spacetime metric Gµ⌫ in the Hamiltonian
decomposition,Gµ⌫ dy µ dy ⌫ =N 2 dt2 + gab (dy a + N a dt)(dy b + N b dt).

Thus, gab is the metric on the spatial bulk leaves of fixed t, Na is the shift vector and N
the lapse function. Since the Lifshitz geometry (2) requires N with a spatial dependence, we
work in the nonprojectable version of HL gravity, with N a full-fledged spacetime-dependent
field. Gauge symmetries are the foliation-preserving di↵eomorphisms of spacetime.
HL gravity may enjoy better short-distance properties than Einstein gravity (if it isdominated at high energies by z > 1 scaling), but here we will follow the “bottom-up”
strategy common in relativistic holography, and work only in the low-energy bulk gravity
approximation. This is equivalent to the large N limit in the dual field theory. In this lowenergy limit, HL gravity is dominated by the most relevant terms compatible with the gauge
symmetries, and its e↵ective action is⇢Z
1↵ 2 ra N ra Np
D
S = 2 dt d x dr gN Kab K ab
K 2 + (R 2⇤) +
.
(4)
2
2
N2
1
(Here Kab = 2N(@t gab ra Nb rb Na ) is the extrinsic curvature of the foliation, R the scalar
curvature of gab , and K = g ab Kab .) The novelty compared to pure Einstein gravity is in thethree couplings , and ↵, which in Einstein gravity are fixed to = = 1 and ↵ = 0.
Note that turning on the ↵ coupling is important for the consistency of nonprojectable HL
gravity [6,7]: Taking the naive ↵ ! 0 limit in (4) would lead to a non-closure of the constraint
algebra.
When ⇤ = 0, the flat spacetime RD+2 is a solution of (4). The propagating graviton
modes consist of the transverse-traceless tensor polarizations with dispersion relation ! 2 =p
k 2 (here k ⌘ ka ka is the magnitude of the spatial momentum), plus an extra scalar graviton
polarization, with dispersion

✓
◆
(1
)
2
!2 =
1+D
1
k2 .
(5)
[1 (D + 1) ]↵2

The requirement of stability and perturbative unitarity around flat spacetime constrains thecouplings to be in the range > 0,
2 D
↵2 
,
(6)
D 1
and
1 or
 1/(D + 1).
(7)
Turning on the cosmological constant ⇤ < 0, we find that the Lifshitz geometry (2) is avacuum solution of HL gravity with low-energy e↵ective action (4), if⇤=(D + z–2–

1)(D + z)
2

(8)

and

2 (z 1).
(9)zThis simple interpretation of Lifshitz spacetimes as vacuum solutions of HL gravity suggeststhat the latter is the natural minimal holographic model of holographic duality for Lifshitztype field theories.
Further evidence for the universality of this minimal model of Lifshitz holography comesfrom the analysis of anisotropic Weyl anomalies in holographic renormalization, initiated in [8](and also later in [9]). Just like their relativistic counterparts, anisotropic Weyl anomalies
contain a lot of universal information about the system, and serve as useful probes of the
duality. Consider again some Lifshitz-type QFT on RD+1 with coordinates (t, xi ) and ageneral background metric
↵2 =

ds2 =

N̂ 2 dt2 + ĝij (dxi + N̂ i dt)(dxj + N̂ j dt).

It is useful to think of this theory as residing at the spacetime boundary r ! 1 of the D + 2
dimensional asymptotically Lifshitz spacetime, with ĝij , N̂i and N̂ being the components of
the appropriately defined (anisotropic conformal class of the) boundary metric [8]. Generally,
we will put hats on quantities defined at the boundary, to distinguish them from their bulk
counterparts. Define anisotropic Weyl transformations, generated by a spacetime dependent(t, xi ):ĝij = 2 ĝij ,N̂i = 2 N̂i ,N̂ = z N̂ .
(10)
These represent a local generalization of the rigid scaling transformations (1). A QFT which
is classically invariant under (10) can develop an anisotropic Weyl anomaly at the quantum
level, with the e↵ective action transforming as (see Appendix C of [8] for details):
ZpSe↵ [ĝij , N̂j , N̂ ] = dt dD x ĝ N̂ (t, xi )A(t, xi ).

The independent terms that can appear in A(t, xi ) are local functionals of the metric, invariant
under foliation preserving di↵eomorphisms. They can be classified by solving a cohomologicalproblem [8] designed to automatically incorporate the Wess-Zumino consistency conditions
on the anomaly. However, the multiplicative coefficients with which these terms contribute to
the anomaly (and which we will refer to as “central charges”) must be calculated for a given
theory on a case-by-case basis. Perhaps the simplest nontrivial case is D = 2 and z = 2. In
this case, the anomaly is [8]!2✓◆i N̂ r̂ N̂
ˆ N̂
1
r̂
i
A = cK K̂ij K̂ ij
K̂ 2 + cV R̂
+
,
(11)
2
N̂ 2
N̂
with two independent central charges, cK and cV . (Here K̂ij , r̂i and R̂ are the extrinsic
curvature, connection and the scalar curvature constructed from ĝij ). As noted in [8], A in(11) takes the form of the Lagrangian for z = 2 conformal HL gravity [2,3] in 2+1 dimensions.

–3–Moreover, while the first term in (11) satisfies the so-called “detailed balance condition,” (i.e.,
it is related to the square of the variation of another local functional W , see [2, 3]), the other
does not.
For QFTs with holographic gravity duals, we can calculate the anomaly by performing
holographic renormalization of the bulk theory [10–12] (see [13–15] for reviews). The relativistic Weyl anomaly was calculated this way in [10]. Holographic renormalization for the
relativistic bulk theory (3) in asymptotically Lifshitz spacetimes was developed and applied
to the anisotropic Weyl anomalies in [8], following the earlier work of [16–19]. This procedure relies substantially on the notion of anisotropic conformal infinity developed in [20]. In
the low-energy gravity approximation, Se↵ [ĝij , N̂i , N̂ ] is calculated by evaluating the on-shell
gravity action with the appropriate fall-o↵ conditions on the metric field as r ! 1. This
on-shell action is divergent due to inifinite volume, and needs to be renormalized. We regulate
it by cutting r o↵ at r✏ = 1/✏, and expand the on-shell action asymptotically to reveal thestructure of its divergences. For the special case of D = z, this expansion gives [8] (modulo
terms that vanish as ✏ ! 0):(D 1)
Z
X L(2n)p
dt dD x ĝ N̂
Le(2D) log ✏ + L(2D) .
2(D n)✏
n=0(The log ✏ term is present only for special values of D and z [8], including the case D = z = 2
of interest here.) The divergent terms are then cancelled by local counterterms, and Se↵ =R
p
dt dD x ĝ N̂ L(2D) .
To calculate these divergent terms, we use the Hamiltonian form of holographic renormalization, as developed for relativistic AdS/CFT in [21, 22] and extended to the asymptoticallyLifshitz case in [8] following [17]. In this formulation, the on-shell bulk action is determined
as a functional of the boundary metric because it satisfies the Hamilton-Jacobi (HJ) equation
for the radial evolution along r. The operator of radial evolution D is given by the generatorof anisotropic dilatations on the boundary, and the HJ equation yields a recursive relation
between the divergent terms L(m) of adjacent scaling dimensions m. One of these recursive
relations implies [8](2D)= 2DL(2D) + Le(2D) .
(12)
DLInterpreted from the boundary point of view, this means that when Le(2D) 6= 0, Se↵ scales
anomalously under the z = D anisotropic Weyl transformations, and Le(2D) is the anisotropic
Weyl anomaly.
For the special case of the relativistic model (3) with D = z = 2, the divergent terms
were calculated in [8], where we obtained for the anisotropic Weyl anomaly
✓
◆
1
1 2
(4)
ij
e
A⌘L =
K̂ij K̂
K̂ .
(13)
16⇡GN
2
Thus, the anomaly in this relativistic model turns out to have cV = 0, or in other words,
satisfies the detailed balance condition. Why is it so? The conclusive answer was found in [8]:–4–The relation implying that the anomaly in the relativistic model (3) should satisfy detailedbalance follows from the holographic recursion relation between the counterterms L(2) andL(4) , with L(2) e↵ectively playing the role of the local functional W (see [8] for details).
Looking for holographic duals of more general QFTs with both central charges independently nonzero is an interesting challenge. Before we embark on this pursuit, we should firstcheck that field theories whose central charges cK and cV are both nonzero indeed exist.
Examples of strongly coupled Lifshitz field theories are very scarce to say the least, but ourpoint can be made by considering the theory of the free z = 2 Lifshitz scalar . When isminimally coupled to background HL gravity,
⇢ ⇣
Z
⌘2
⇣
⌘2
p
1
2
S = dt d x ĝ@tN̂ i r̂iN̂ ˆ,N̂this theory is classically invariant under (10) (with= 0), but develops an anisotropic
Weyl anomaly at the quantum level. This anomaly was calculated in [9], and it turns out tohave cV = 0. One could perhaps speculate that cV = 0 might be a universal property of allconsistent QFTs, hence eliminating the need for finding gravity duals with cV 6= 0. A simple
counterexample comes from coupling to background gravity non-minimally, adding
(
)2
Z
p
N̂
ri N̂ ri N̂
2
D2e
dt d x ĝ N̂ R̂
+
2
N̂
N̂

to S . Even with this non-minimal coupling, this theory stays classically invariant under the
anisotropic Weyl transformations (again with= 0), and develops a quantum anomaly.
We calculated this anisotropic Weyl anomaly using the ⇣-function regularization, and foundcK = 1/(32⇡) and cV = e2 /(8⇡).
Having demonstrated the existence of QFTs with cV 6= 0, we can now ask how to reproduce this second central charge in a holographic gravity dual. One could look for relativistic
models more complicated than (3). Instead, we will show that minimal HL gravity alreadyaccounts for both of the independent central charges cK and cV in the anisotropic Weyl
anomaly. In order to show that, we have performed holographic renormalization of Lifshitz
spacetimes in pure HL gravity. Since the technicalities are quite involved (as they were in
the relativistic model (3) studied in [8]), here we only present our main results; all technicaldetails will appear in [23].
We find modified recursion relations for the divergent terms in the regulated action. In
the special case D = z = 2, we solved these recursion relations and found that the logarithmic
term Le(4) is equal to!2✓◆i N̂ r̂ N̂
ˆ N̂
1
1
r̂
i
(4)
ij
2
Le = 2 K̂ij K̂
K̂ +
R̂
+
.
2
2
482
N̂ 2
N̂
This is the anisotropic Weyl anomaly in our minimal model of Lifshitz holography withvacuum HL gravity. It is indeed of the most general form, with the two independent central–5–charges given in terms of two low-energy couplings in minimal HL gravity: cK = 1/(22 ) andcV = /(482 ).
The remaining coupling does not appear in the anomaly, but it still plays an important
physical role. Just as in the case of flat spacetime, enters into the dispersion relation of
the extra polarization of the graviton in the bulk. For example in the radial gauge gij = 0,
grr = 1/r2 , the extra graviton mode is found as the linearized fluctuation of the radial
component of the shift vector, Nr = /r. Returning to the case of general D and z, thelinearized equations of motion imply the asymptotic behavior near infinity (r) ⇠ rD± , with
(
)
r
1
4D(1
z)
D± =
z D ± (z + D)2 +
.
(14)
2
1
Standard rules of holographic duality will map D± to the scaling dimensions ± of theoperator dual to the extra graviton. Unitarity of the dual field theory requires that thescaling dimensions be real, implying (for z > 1)
1

or



U ⌘

(D

z)2 + 4D.
(D + z)2(15)

This unitarity bound represents an intriguing analog of the Breitenlohner-Freedman boundfamiliar from relativistic holography: In HL gravity, the unitarity bound (15) allows thecoupling to dip into the region between 1/(D + 1) and 1, which according to (7) would
be forbidden around flat spacetime. In the particularly interesting case of D = z, we get 1/D.
U = 1/D, which opens up the previously forbidden regime 1/(D + 1) 
Now that we have seen that HL gravity provides candidate holographic duals for QFTs
with anisotropic Lifshitz scaling, is it possible to apply HL gravity also to QFTs with isotropicz = 1 scaling? Interestingly, the limit z ! 1 corresponds to ↵ ! 0, the “unhealthy reduction” [7] of nonprojectable HL gravity, and may therefore be difficult to make sense of. Thisis perhaps to be expected: z = 1 QFTs with such gravity duals would likely exhibit isotropic
dilatation symmetry without full relativistic conformal symmetry, a phenomenon whose examples are few and far between. Further study of our holographic duality in the ↵ ! 0 limit
may shed new light on this rare class of QFTs.
Finally, throughout this paper we have used the e↵ective low-energy limit of HL gravity,
dominated by the terms of the lowest dimension in the action. We have been agnostic about
how the model is completed at high energies. This completion may come from additional
degrees of freedom, perhaps via an embedding into string theory; or it can be via a selfcompletion of HL gravity, due to highly anisotropic scaling at short distances. This latter
possibility would be particularly interesting, as it could open a new door away from the large
N limit and small bulk curvature. As this paper was being finalized, complementary results
about another form of nonrelativistic holography with HL gravity were presented in [24, 25].
Our results, and those of [24, 25], thus provide further evidence for the picture proposed
originally in [8], that the natural arena for nonrelativistic holography is nonrelativistic HL
gravity. It remains to be seen whether – as suggested in [8] – the nonrelativistic field theories–6–

whose holographic duals happen to be relativistic indeed represent only a minority among all
theories with gravity duals.

Acknowledgments
We wish to thank Omid Saremi and Kostas Skenderis for useful discussions. The resultspresented in this paper were reported by one of the authors (PH) at the “32nd Winter School
on Geometry and Physics,” Srnı́, Šumava, Czech Republic (January 2012), the workshops“Exploring the Quantum Spacetime,” Bad Honnef, Germany (March 2012) and “Quantum
Gravity in Paris,” Paris, France (March 2012), the “6th International School on Field Theory and Gravitation,” Petrópolis, Brazil (April 2012), “The Conformal Universe” workshop,
Perimeter Institute, Waterloo, Canada (May 2012), in a plenary talk at the “13th Marcel
Grossmann Meeting MG13,” Stockholm, Sweden (July 2012), and at the “2nd BCTP Summit at Lake Tahoe,” Glenbrook, Nevada (September 2012); and by one of us (CMT) at the
University of Washington (April 2012). PH and CMT wish to thank the organizers and hostsof these events for their wonderful hospitality. This work has been supported in part by NSF
Grants PHY-0855653 and PHY-1214644, DOE Grant DE-AC02-05CH11231, and BCTP.

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