Holographic Renormalization Group with Gravitational Chern-Simons Term
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Holographic Renormalization Group with Gravitational Chern-Simons Term
Takahiro Nishinaka( Osaka U.)
(Collaborators: K. Hotta, Y. Hyakutake, T. Kubota and H. Tanida )
( arXiv: 0906.1255 [hep-th] )
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Introduction
“C-theorem“ is one of the most interesting features of 2-dim QFT.
c- function : # degrees of freedom
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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
![Page 4: Holographic Renormalization Group with Gravitational Chern-Simons Term](https://reader034.fdocuments.us/reader034/viewer/2022052510/56812bc7550346895d90129c/html5/thumbnails/4.jpg)
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
![Page 5: Holographic Renormalization Group with Gravitational Chern-Simons Term](https://reader034.fdocuments.us/reader034/viewer/2022052510/56812bc7550346895d90129c/html5/thumbnails/5.jpg)
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
![Page 6: Holographic Renormalization Group with Gravitational Chern-Simons Term](https://reader034.fdocuments.us/reader034/viewer/2022052510/56812bc7550346895d90129c/html5/thumbnails/6.jpg)
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.
![Page 7: Holographic Renormalization Group with Gravitational Chern-Simons Term](https://reader034.fdocuments.us/reader034/viewer/2022052510/56812bc7550346895d90129c/html5/thumbnails/7.jpg)
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
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Parity-Violating 2-dim QFT
c-functions
: length scale
At the fixed point,coincide with two central charges.
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Parity-Violating 2-dim QFT
c-functions
: length scale
At the fixed point,coincide with two central charges.
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Parity-Violating 2-dim QFT
Weyl anomaly
c-functions
: length scale
At the fixed point,coincide with two central charges.
![Page 11: Holographic Renormalization Group with Gravitational Chern-Simons Term](https://reader034.fdocuments.us/reader034/viewer/2022052510/56812bc7550346895d90129c/html5/thumbnails/11.jpg)
Parity-Violating 2-dim QFT
Weyl anomaly
Gravitational anomaly
c-functions
: length scale
At the fixed point,coincide with two central charges.
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Parity-Violating 2-dim QFT
Weyl anomaly
Gravitational anomaly
Bardeen-Zumino polynomial (making energy-momentum tensor covariant)
c-functions
: length scale
At the fixed point,coincide with two central charges.
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Holographic Renormalization Group
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Holographic Renormalization Group
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This is a dual description of the RG-flow of 2-dimensional QFT.
UV
IR
Holographic Renormalization Group
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TMG + Scalar scalar
gravitational Chern-Simons term
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TMG + Scalar scalar
gravitational Chern-Simons term
ADM decomposition
We here decompose metric into the radial direction and 2-dim spacetime.
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TMG + Scalar
: auxiliary fields
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TMG + Scalar
Since the action contains the third derivative of , we treat as independent dynamical variables.
: auxiliary fields
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TMG + Scalar
Since the action contains the third derivative of , we treat as independent dynamical variables.
: auxiliary fields
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TMG + Scalar
: auxiliary fields
Since the action contains the third derivative of , we treat as independent dynamical variables.
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TMG + Scalar
Since the action contains the third derivative of , we treat as independent dynamical variables. Momenta conjugate to them are
: auxiliary fields
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Hamilton-Jacobi Equation
Hamiltonian is given by constraints:
contain
and also
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Hamilton-Jacobi Equation
Hamiltonian is given by constraints:
Constraints from path integration over auxiliary fields are
contain
and also
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Hamilton-Jacobi Equation
Hamiltonian is given by constraints:
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 .
contain
and also
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Hamilton-Jacobi Equation
First, path integration over leads to
from which we can remove .
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Hamilton-Jacobi Equation
First, path integration over leads to
Moreover, by using a classical action, we can also remove from Hamiltonian.
where the classical solution is substituted into .
from which we can remove .
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Hamilton-Jacobi Equation
First, path integration over leads to
Moreover, by using a classical action, we can also remove from Hamiltonian.
where the classical solution is substituted into . Then are
from which we can remove .
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Holographic Renormalization
The bulk action is a functional of boundary conditions .
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Holographic Renormalization
The bulk action is a functional of boundary conditions .
We divide according to weight. includes only terms with
weight .
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Holographic Renormalization
The bulk action is a functional of boundary conditions .
We divide according to weight. includes only terms with
weight . The weight is assigned as follows:
[Fukuma, Matsuura, Sakai]
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Holographic Renormalization
The bulk action is a functional of boundary conditions .
We divide according to weight. includes only terms with
weight . The weight is assigned as follows:
We regard as a quantum action of dual field theory, which might contain non-local terms.
[Fukuma, Matsuura, Sakai]
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We now study the physical meanings of , or
by comparing weights of both sides.
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Hamiltonian Constraint and Weyl Anomaly
From terms in , we can determine weight-zero counterterms :
where
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Hamiltonian Constraint and Weyl Anomaly
From terms in , we can obtain the RG equation in 2-dim:
: constant
From terms in , we can determine weight-zero counterterms :
where
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Hamiltonian Constraint and Weyl Anomaly
From terms in , we can obtain the RG equation in 2-dim:
And we can also read off the Weyl anomaly in the 2-dim QFT:
: constant
From terms in , we can determine weight-zero counterterms :
where
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Hamiltonian Constraint and Weyl Anomaly
From terms in , we can obtain the RG equation in 2-dim:
And we can also read off the Weyl anomaly in the 2-dim QFT:
: constant
cf.) In 2-dim,
From terms in , we can determine weight-zero counterterms :
where
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Momentum Constraint and Gravitational AnomalyFrom weight three terms of the second constraint , we can read off the gravitational anomaly in the 2-dim QFT.
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Momentum Constraint and Gravitational AnomalyFrom weight three terms of the second constraint , we can read off the gravitational anomaly in the 2-dim QFT.
In pure gravity case, the RHS is zero which meansenergy-momentum conservation.
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Momentum Constraint and Gravitational AnomalyFrom weight three terms of the second constraint , we can read off the gravitational anomaly in the 2-dim QFT.
cf.) In 2-dim, In pure gravity case, the RHS is zero which meansenergy-momentum conservation.
![Page 41: Holographic Renormalization Group with Gravitational Chern-Simons Term](https://reader034.fdocuments.us/reader034/viewer/2022052510/56812bc7550346895d90129c/html5/thumbnails/41.jpg)
Momentum Constraint and Gravitational AnomalyFrom weight three terms of the second constraint , we can read off the gravitational anomaly in the 2-dim QFT.
cf.) In 2-dim,
Bardeen-Zumino term: non-covariant terms which make energy-momentum tensor general covariant.
In pure gravity case, the RHS is zero which meansenergy-momentum conservation.
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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 !!
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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 left-right 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.
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That‘s all for my presentation.
Thank you very much.