Holographic Renormalization Group with
Gravitational Chern-Simons Term
( arXiv: 0906.1255 [hep-th] )
Takahiro Nishinaka
( Osaka U.)
(Collaborators: K. Hotta, Y. Hyakutake, T. Kubota and H. Tanida )
Introduction
 “C-theorem“ is one of the most interesting features of 2-dim QFT.
c- function :
# degrees of freedom
Introduction
 “C-theorem“ is one of the most interesting features of 2-dim QFT.
c- function :
# degrees of freedom
monotonically decreasing along the renormalization group flow
Introduction
 “C-theorem“ is one of the most interesting features of 2-dim QFT.
c- function :
# degrees of freedom
monotonically decreasing along the renormalization group flow
 By virtue of holography, we can analyze this from 3-dim gravity.
pure gravity
+
scalar
Introduction
 “C-theorem“ is one of the most interesting features of 2-dim QFT.
c- function :
# degrees of freedom
monotonically decreasing along the renormalization group flow
 By virtue of holography, we can analyze this from 3-dim gravity.
pure gravity
+
scalar
Weyl anomaly calculation from gravity
Introduction
 “C-theorem“ is one of the most interesting features of 2-dim QFT.
c- function :
# degrees of freedom
monotonically decreasing along the renormalization group flow
 By virtue of holography, we can analyze this from 3-dim gravity.
pure gravity
+
scalar
Weyl anomaly calculation from gravity
 C-theorem is, however, known to be satisfied even when
Now
is constant along the renormalization group.
.
Introduction
 “C-theorem“ is one of the most interesting features of 2-dim QFT.
c- function :
# degrees of freedom
monotonically decreasing along the renormalization group flow
 By virtue of holography, we can analyze this from 3-dim gravity.
pure gravity
+
scalar
Weyl anomaly calculation from gravity
 C-theorem is, however, known to be satisfied even when
Now
is constant along the renormalization group.
As a dual gravity set-up, we consider
Topologically Massive Gravity (TMG)
+
scalar
.
Parity-Violating 2-dim QFT
c-functions
: length scale
At the fixed point,
coincide with two central charges.
Parity-Violating 2-dim QFT
c-functions
: length scale
At the fixed point,
coincide with two central charges.
Parity-Violating 2-dim QFT
c-functions
: length scale
At the fixed point,
coincide with two central charges.
Weyl anomaly
Parity-Violating 2-dim QFT
c-functions
: length scale
At the fixed point,
coincide with two central charges.
Weyl anomaly
Gravitational anomaly
Parity-Violating 2-dim QFT
c-functions
: length scale
At the fixed point,
coincide with two central charges.
Weyl anomaly
Gravitational anomaly
Bardeen-Zumino polynomial (making energy-momentum tensor covariant)
Holographic Renormalization Group
Holographic Renormalization Group
Holographic Renormalization Group
UV
IR
This is a dual description of the RG-flow of 2-dimensional QFT.
TMG + Scalar
scalar
gravitational Chern-Simons term
TMG + Scalar
scalar
gravitational Chern-Simons term
ADM decomposition
We here decompose metric into the radial direction and 2-dim spacetime.
TMG + Scalar
: auxiliary fields
TMG + Scalar
: auxiliary fields
 Since the action contains the third derivative of
as independent dynamical variables.
, we treat
TMG + Scalar
: auxiliary fields
 Since the action contains the third derivative of
as independent dynamical variables.
, we treat
TMG + Scalar
: auxiliary fields
 Since the action contains the third derivative of
as independent dynamical variables.
, we treat
TMG + Scalar
: auxiliary fields
 Since the action contains the third derivative of
, we treat
as independent dynamical variables. Momenta conjugate to them are
Hamilton-Jacobi Equation
Hamiltonian is given by constraints:
contain
and also
Hamilton-Jacobi Equation
Hamiltonian is given by constraints:
contain
and also
Constraints from path integration over auxiliary fields
are
Hamilton-Jacobi Equation
Hamiltonian is given by constraints:
contain
and also
Constraints from path integration over auxiliary fields
are
In order to see the physical meanings of these constraints, we have to
express
only in terms of the boundary conditions
.
Hamilton-Jacobi Equation
First, path integration over
from which we can remove
leads to
.
Hamilton-Jacobi Equation
First, path integration over
from which we can remove
leads to
.
Moreover, by using a classical action, we can also remove
from Hamiltonian.
where the classical solution is substituted into
.
Hamilton-Jacobi Equation
First, path integration over
from which we can remove
leads to
.
Moreover, by using a classical action, we can also remove
from Hamiltonian.
where the classical solution is substituted into
. Then
are
Holographic Renormalization
The bulk action
is a functional of boundary conditions
.
Holographic Renormalization
The bulk action
We divide
weight
.
is a functional of boundary conditions
according to weight.
includes only terms with
.
Holographic Renormalization
The bulk action
We divide
weight
is a functional of boundary conditions
according to weight.
.
includes only terms with
. The weight is assigned as follows:
[Fukuma, Matsuura, Sakai]
Holographic Renormalization
The bulk action
We divide
weight
is a functional of boundary conditions
according to weight.
.
includes only terms with
. The weight is assigned as follows:
[Fukuma, Matsuura, Sakai]
We regard
as a quantum action of dual field theory, which might
contain non-local terms.
We now study the physical meanings of
by comparing weights of both sides.
, or
Hamiltonian Constraint and Weyl Anomaly
From
terms in
counterterms :
, we can determine weight-zero
where
Hamiltonian Constraint and Weyl Anomaly
From
terms in
counterterms :
, we can determine weight-zero
where
From
terms in
, we can obtain the RG equation in 2-dim:
: constant
Hamiltonian Constraint and Weyl Anomaly
From
terms in
counterterms :
, we can determine weight-zero
where
From
terms in
, we can obtain the RG equation in 2-dim:
: constant
And we can also read off the Weyl anomaly in the 2-dim QFT:
Hamiltonian Constraint and Weyl Anomaly
From
terms in
counterterms :
, we can determine weight-zero
where
From
terms in
, we can obtain the RG equation in 2-dim:
: constant
And we can also read off the Weyl anomaly in the 2-dim QFT:
cf.) In 2-dim,
Momentum Constraint and Gravitational Anomaly
From weight three terms of the second constraint
read off the gravitational anomaly in the 2-dim QFT.
, we can
Momentum Constraint and Gravitational Anomaly
From weight three terms of the second constraint
read off the gravitational anomaly in the 2-dim QFT.
In pure gravity case,
the RHS is zero which means
energy-momentum conservation.
, we can
Momentum Constraint and Gravitational Anomaly
From weight three terms of the second constraint
read off the gravitational anomaly in the 2-dim QFT.
In pure gravity case,
the RHS is zero which means
energy-momentum conservation.
cf.) In 2-dim,
, we can
Momentum Constraint and Gravitational Anomaly
From weight three terms of the second constraint
read off the gravitational anomaly in the 2-dim QFT.
In pure gravity case,
the RHS is zero which means
energy-momentum conservation.
, we can
cf.) In 2-dim,
Bardeen-Zumino term: non-covariant terms which make
energy-momentum tensor general covariant.
Holographic c-functions
We can define left-right asymmetric c-functions as follows:
where
depends on the radial coordinate and
is constant along the renormalization group flow !!
Summary
 We study Topologically Massive Gravity (TMG) + scalar system in 3
dimensions as a dual description of the RG-flow of 2-dimensional
QFT.
 Due to the gravitational Chern-Simons coupling, We can obtain leftright asymmetric c-functions holographically.

is constant along the renormalization group flow, which is
consistent with the property of 2-dim QFT.
 The Bardeen-Zumino polynomial is also seen in gravity side.
That‘s all for my presentation.
Thank you very much.