JHEP04(2018)134
the dual one point functions was made e x pl i c i t in [23], where it was shown how to recast
these observables in term of horizon quantities by explicitly solving the bulk eq uat i ons of
motion. Furthermore it was recently showed by various authors [
24, 25] that a part of this
contri but i on can be explained through the matching of the global gravitational anomalies.
In this work we connect t he aforementioned two lines of research, i.e. holographic con-
served charges and anomalous transport. In ord er to do so, we show how the membrane
paradigm can be extended to bulk Chern-Simons theorie s by suitably modifying the defini-
tions of the membrane currents and stress tensor. To thi s end, we will need t o examine the
constraint equations on arbitr ary hypersur fac es of constant radial coordinate to properly
define the membrane observables and show that these match wi t h the conserved quantities
given by the Komar charges.
The Wald construction is slightly technical in this case due to the noncovariant nature
of the Chern-Simons action for the bulk theory. It was generalized to these actions in
various works [
26–28]. Other authors have used this form al i sm to perform a near horizon
analysis of the entropy current [
22]. An argument closer in spirit to the work of Iqbal and
Liu was given by [
29, 30], where the conserved quantity is constructed for the U(1) currents
in an anomalous theory. They however lack an explicit argument for energy fluctuations.
In this note we aim to generalize and unify the various insights above. In particu-
lar, we include the energy current transport and we explicitly r e l ate it to the membrane
paradigm by the construction of t he membrane currents for the anomalous theory. This
last constructi on is not trivial, as bulk gravitational Chern-Simons terms give rise to higher
derivative theories for which the Cauchy problem is in general not well defined. This gives
rise to s ubt l et i e s in defining the membrane stress tensor away from the conformal bound-
ary. Furthermore, it is well known that in anomalous theories the current operators have
no unique definition but one can defi ne differe nt operators depending on the properties
which are desired e.g. consistent and covariant currents. We shed light on the role of these
different operators from the point of view of the bulk dynamics.
We will then use the appropriate conserved charges to match the membrane res ul ts
to the boundary observables, highlighting the d y nami c al mechanisms that give rise to
the anomalous contributions from our perspective. In particular, we will show that the
gravit at i onal contributions to anomalous transport come from horizon extrinsic Chern-
Simons currents which are dynamically generated along t he radial directi on.
The work is organize d as follows: in s ec t i on
2 we review the extraction of th e Wald
charges in Einstein-Max well theory, discuss i ng their link to the bulk constraint equations,
and t he construction of the membr ane current and stress-tensor. In section
3 we use the
constraint equations of the bulk theory to propose a definition of membrane current and
stress-tensor. We give some consistency checks for this proposal. In section
4 we review the
generalized Wald construction as done by [
27, 28] and show that the continuity equations
for the resulting close d − 1 forms can be understood as RG equations for the previously
defined membrane currents. We integrate them to recover the known anomalous transport
coefficients and interpret them in the spirit of membrane paradigm.
As the topic of anomalous transport in holography has al r eady been widely studied let
us add a few comments on the differences and similarities with previous works :
– 4 –
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