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JHEP04(2016)112

Published for SISSA by Springer

Received: February 1, 2016

Accepted: April 11, 2016

Published: April 19, 2016

Membrane paradigm, gravitational Θ-term and

gauge/gravity duality

Willy Fischler

a,b

and Sandipan Kundu

Theory Group, Department of Physics, University of Texas,

Austin, TX 78712, U.S.A.

Texas Cosmology Center, University of Texas,

Austin, TX 78712, U.S.A.

Department of Physics, Cornell University,

Ithaca, New York, 14853, U.S.A.

E-mail: fischler@physics.utexas.edu, kundu@cornell.edu

Abstract: Following the membrane paradigm, we explore the eﬀect of the gravitational

Θ-term on the behavior of the stretched horizon of a black hole in (3 + 1)-dimensions. We

reformulate the membrane paradigm from a quantum path-integral point of view where

we interpret the macroscopic properties of the horizon as eﬀects of integrating out the

region inside the horizon. The gravitational Θ-term is a total derivative, however, using

our framework we show that this term aﬀects the transport properties of the horizon.

In particular, the horizon acquires a third order parity violating, dimensionless transport

coeﬃcient which aﬀects the way localized perturbations scramble on the horizon. Then we

consider a large-N gauge theory in (2 + 1)-dimensions which is dual to an asymptotically

AdS background in (3 + 1)-dimensional spacetime to show that the Θ-term induces a non-

trivial contact term in the energy-momentum tensor of the dual theory. As a consequence,

the dual gauge theory in the presence of the Θ-term acquires the same third order parity

violating transport coeﬃcient.

Keywords: AdS-CFT Correspondence, Black Holes, Classical Theories of Gravity, CP

violation

ArXiv ePrint: 1512.01238

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP04(2016)112

JHEP04(2016)112

Historically the membrane paradigm [3, 4] has also been successful at providing us

with a powerful framework to study macroscopic properties of black hole horizons. In

astrophysics the membrane paradigm has been used extensively as an eﬃcient compu-

tational tool to study phenomena in the vicinity of black holes (see [4–9] and references

therein). The membrane paradigm has also been able to provide crucial hints about details

of the microscopic physics of horizons. In particular, the membrane paradigm predicts that

black hole horizons are the fastest scramblers in nature. Fast-scrambling strongly indicates

that the microscopic description of scrambling of information on static horizons must in-

volve non-local degrees of freedom [10, 11]. In this paper, we will try to understand the

membrane paradigm from a quantum path-integral point of view. We will interpret the

macroscopic properties of the horizon as eﬀects of integrating out the region inside the

horizon. The semi-classical approximation of this path-integral approach is equivalent to

the action formulation [12] of the conventional membrane paradigm.

We are mainly interested in ﬁguring out how total derivative terms can aﬀect the

macroscopic properties of black hole horizons. Total derivative terms do not aﬀect the

classical equations of motion and hence do not contribute even in perturbative quantum

ﬁeld theory. However, it is well known that total derivative terms can have physical ef-

fects, e.g. Lorentz and gauge invariance of Quantum chromodynamics (QCD) allow for a

CP-violating topological θ

QCD

term which contributes to the electric dipole moment of

neutrons [13]. Similarly, the electrodynamics θ-term is also a total derivative, therefore,

does not contribute for perturbative quantum electrodynamics (QED). However, in the

presence of the electrodynamics θ-angle a black hole horizon behaves as a Hall conduc-

tor, for an observer hovering outside [14]. As a consequence, the electrodynamics θ-angle

aﬀects the way localized perturbations, created on the stretched horizon by dropping a

charged particle, fast scramble on the horizon [14]. Later it was also shown that in-falling

electric charges produce a non-trivial Berry phase in the QED wave function which can

have physical eﬀects in the early universe [15].

Another example comes from gravity in (3 + 1)-dimensions, where the topological

Gauss-Bonnet term contributes a correction term to the entropy of a black hole which

is proportional to the Euler number of the horizon. One can show that this correction

term violates the second law of black hole thermodynamics and hence should be zero [16–

19]. In (3 + 1)-dimensions, there exists another total derivative term, a parity violating

gravitational Θ-term

x 

µναβ

σµν

ταβ

In this paper, we explore the eﬀect of this Θ-term on black hole horizons. The membrane

paradigm tells us that for an outside observer a black hole horizon eﬀectively behaves

like a viscous Newtonian ﬂuid. Using our framework, we will show that the gravitational

Θ-term aﬀects the transport properties of the horizon ﬂuid, in particular, the horizon

ﬂuid acquires a third order parity violating, dimensionless transport coeﬃcient, which we

will call ϑ. This indicates that the Θ-term will aﬀect the way perturbations scramble

on the horizon. Speciﬁcally, we can perform a thought experiment, in which an outside

observer drops a massive particle onto the black hole and watches how the perturbation

– 2 –

JHEP04(2016)112

scrambles on the black hole horizon. We will argue that the gravitational Θ-term, similar

to the electrodynamics θ-term, will also introduce vortices without changing the scrambling

time. This strongly suggests that in a sensible theory of quantum gravity the Θ-term will

play an important role, a claim that we will show is also supported by the AdS/CFT

correspondence.

The membrane paradigm has become even more relevant with the emergence of holog-

raphy [1, 2], a remarkable idea that connects two cornerstones of theoretical physics: quan-

tum gravity and gauge theory. The AdS/CFT correspondence [2], which is a concrete

realization of this idea of holography, has successfully provided us with theoretical control

over a large class of strongly interacting ﬁeld theories [2, 20–22]. This duality enables us to

compute observables of certain large-N gauge theories in d-dimensions by performing some

classical gravity calculations in (d + 1)-dimensions. Gravity duals of these ﬁeld theories at

ﬁnite temperature contain black holes with horizons. It has been shown that there is some

connection between the low frequency limit of linear response of a strongly coupled quan-

tum ﬁeld theory and the membrane paradigm ﬂuid on the black hole horizon of the dual

gravity theory [23–27]. In this paper, we will consider a large-N gauge theory in (2 + 1)-

dimensions which is dual to a gravity theory in (3 + 1)-dimensions with the gravitational

Θ-term and ﬁgure out the eﬀect of the parity violating Θ-term on the dual ﬁeld theory.

A reasonable guess is that the boundary theory, similar to the membrane paradigm ﬂuid,

will acquire the same third order parity violating transport coeﬃcient ϑ. We will conﬁrm

this guess by performing an explicit computation.

It was argued in [28] that the two-point function of the energy-momentum tensor in a

(2 + 1)-dimensional conformal ﬁeld theory can have a non-trivial contact term

(x)T

(0)i = −i

192π

h

iml

∂



∂

− ∂



+ (i ↔ j)



+ (m ↔ n)

(x) .

It is possible to shift κ

by an integer by adding a gravitational Chern-Simons counterterm

to the UV-Lagrangian and hence the integer part of κ

is scheme-dependent. On the other

hand, the fractional part κ

mod 1 does not depend on the short distance physics and hence

it is a meaningful physical observable in (2 + 1)-dimensional conformal ﬁeld theory [28].

We will argue that a gravity theory in AdS

(3+1)

with the gravitational Θ-term is dual to a

conformal ﬁeld theory with non-vanishing κ

, in particular

96π

= Θ

which also suggests that only a fractional part of the Θ-term is a well-deﬁned observable.

This also suggests that Θ, in our normalization is not an angle. However, one can work in the normal-

ization in which the gravitational Θ-term takes the form

1536π

x 

µναβ

σµν

ταβ

In that case, adding an integer to κ

changes the new Θ by an integer times 2π and hence in the above

normalization Θ is an angle.

– 3 –

JHEP04(2016)112

The contact term κ

is also related to the transport coeﬃcient ϑ, which to our knowl-

edge has never been studied before. It is a parity violating third order

transport coeﬃcient

in (2+1)-dimensions and hence forbidden in a parity-invariant theory. Under a small metric

perturbation γ

around ﬂat Minkwoski metric, it contributes to the energy-momentum

tensor in the following way:

= −T

= −ϑ

∂

∂t

, T

= T



∂

∂t

−

∂

∂t



and hence ϑ contributes to the retarded Green’s function of the energy-momentum tensor

in order ω

12,11−22

(ω,

k → 0) = −2iϑω

ϑ is dimensionless and it does not aﬀect the trace of the energy-momentum tensor. In

(2 + 1)-dimensional hydrodynamics, the Hall viscosity is another parity violating eﬀect

that appears in the ﬁrst order in derivative expansion. The Hall viscosity has been studied

extensively for both relativistic [30] and non-relativistic systems [31–33]. We believe that

ϑ is a third order cousin of Hall viscosity and hence it is also an example of Berry-like

transport [34]. We will show that for a holographic theory dual to asymptotically AdS

spacetime in (3 + 1)-dimensions: ϑ = Θ = κ

/96π. We will also speculate on the possible

covariant structure of the ϑ contribution to the energy-momentum tensor.

The rest of the paper is organized as follows. We start with a discussion of the mem-

brane paradigm in section 2. In section 3, we review the membrane paradigm for the

Einstein gravity. Then in section 4, we introduce gravitational Θ-term and discuss its

eﬀect on the stretched horizon. In section 5, we discuss the eﬀect of the Θ-term in the

context of the AdS/CFT correspondence and make some comments on the ϑ-transport in

section 6. Finally, we conclude in section 7. Some technical details have been relegated to

appendices A and B. For readers only interested in the eﬀect of the Θ-term in the context

of the AdS/CFT correspondence, it is suﬃcient to read sections 5 and 6.

2 Integrating out inside: membrane paradigm

The membrane paradigm provides a simple formalism to study macroscopic properties of

horizons by replacing the true mathematical horizon by a stretched horizon, an eﬀective

time-like membrane located roughly one Planck length away from the true horizon. Finite-

ness of the black hole entropy suggests that between the actual black hole horizon and the

stretched horizon, the eﬀective number of degrees of freedom should be vanishingly small.

So, it is more natural as well as convenient to replace the true mathematical horizon by a

stretched horizon.

Predictions of the membrane paradigm are generally considered to be robust since they

depend on some very general assumptions:

Little is known about third-order transport coeﬃcients in any dimensions. Very recently, third order

hydrodynamics for neutral ﬂuids has been studied in (3 + 1)-dimensions [29].

– 4 –

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