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] )

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

Weyl anomaly

c-functions

: length scale

At the fixed point,coincide with two central charges.

Parity-Violating 2-dim QFT

Weyl anomaly

Gravitational anomaly

c-functions

: length scale

At the fixed point,coincide with two central charges.

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.

Holographic Renormalization Group

Holographic Renormalization Group

This is a dual description of the RG-flow of 2-dimensional QFT.

UV

IR

Holographic Renormalization Group

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

Since the action contains the third derivative of , we treat as independent dynamical variables.

: auxiliary fields

TMG + Scalar

Since the action contains the third derivative of , we treat as independent dynamical variables.

: auxiliary fields

TMG + Scalar

: auxiliary fields

Since the action contains the third derivative of , we treat as independent dynamical variables.

TMG + Scalar

Since the action contains the third derivative of , we treat as independent dynamical variables. Momenta conjugate to them are

: auxiliary fields

Hamilton-Jacobi Equation

Hamiltonian is given by constraints:

contain

and also

Hamilton-Jacobi Equation

Hamiltonian is given by constraints:

Constraints from path integration over auxiliary fields are

contain

and also

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

Hamilton-Jacobi Equation

First, path integration over leads to

from which we can remove .

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 .

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 .

Holographic Renormalization

The bulk action is a functional of boundary conditions .

Holographic Renormalization

The bulk action is a functional of boundary conditions .

We divide according to weight. includes only terms with

weight .

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]

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]

We now study the physical meanings of , or

by comparing weights of both sides.

Hamiltonian Constraint and Weyl Anomaly

From terms in , we can determine weight-zero counterterms :

where

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

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

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

Momentum Constraint and Gravitational AnomalyFrom weight three terms of the second constraint , we can read off the gravitational anomaly in the 2-dim QFT.

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.

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.

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.

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 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.

That‘s all for my presentation.

Thank you very much.