Transcript A New Class of Conformal Field Theories with Anomalous

The nonperturbative analyses for
lower dimensional non-linear sigma
models
Etsuko Itou (Kanazawa University)
1.Introduction
2.The WRG equation for NLσM
3.Fixed points with U(N) symmetry
4.3-dimensional case
5. Summary
1. Introduction
We consider the Wilsonian effective action which has
derivative interactions.
It corresponds to the non-linear sigma model action,
so we compare the results with the perturbative one.
It corresponds to next-to-leading order approximation in
derivative expansion.
Local potential term
Local potential approximation :
Non-linear sigma model
K.Aoki Int.J.Mod.Phys. B14 (2000) 1249
T.R.Morris Int.J.Mod.Phys. A9 (1994) 2411
The point of view of non-linear sigma model:
Two-dimensional case
●In perturbative analysis, the 1-loop b
function for 2-dimensional non-linear sigma
model proportional to Ricci tensor of target
spaces. ⇒Ricci Flat
Is there the other fixed point?
The perturbative results
Alvarez-Gaume, Freedman and Mukhi Ann. of Phys. 134 (1982) 392
Three-dimensional case
●The 3-dimensional non-linear sigma models
are nonrenormalizable within the perturbative
method. We need some nonperturbative
renormalization methods.
WRG approach
Large-N expansion
Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （2000）1283
2.The WRG equation for NLσM
The Euclidean path integral is
K.Aoki Int.J.Mod.Phys. B14 (2000) 1249
The Wilsonian effective action has infinite number of interaction terms.
The WRG equation (Wegner-Houghton equation) describes the variation
of effective action when energy scale L is changed to L(dt)=L exp[-dt] .
To obtain the WRG eq. , we integrate shell mode.
only
The Wilsonian RG equation is written as follow:
Field rescaling effects to normalize kinetic terms.
We use the sharp cutoff equation. It corresponds to the sharp
cutoff limit of Polchinski equation at least local potential level.
Approximation method:
Symmetry and Derivative expansion
Consider a single real scalar field theory that is invariant under
We expand the most generic action as
In this work, we expand the action up to second order
in derivative and assume it =2 supersymmetry.
D=2 (3) N =2 supersymmetric
non linear sigma model
i=1～N：N is the dimensions of target spaces
Where K is Kaehler potential and F is chiral superfield.
We expand the action around the scalar fields.
where
: the metric of target spaces
From equation of motion, the auxiliary filed F can be vanished.
Considering only Kaehler potential term corresponds to second order
to derivative for scalar field.
There is not local potential term.
The WRG equation for non
linear sigma model
Consider the bosonic part of the action.
The second term of the right hand side vanishes in this
approximation O( ) .
The first term of the right hand side
From the bosonic part of the action
From the fermionic kinetic term
Non derivative term is cancelled.
Finally, we obtain the WRG eq. for bosonic part of the action as follow:
The b function for the Kaehler metric is
The perturbative results
Alvarez-Gaume, Freedman and Mukhi Ann. of Phys. 134 (1982) 392
3. Fixed points with U(N) symmetry
The perturbative βfunction follows the
Ricci-flat target manifolds.
Ricci-flat
We derive the action of the conformal field theory corresponding to
the fixed point of the b function.
To simplify, we assume U(N) symmetry for Kaehler potential.
where
The function f(x) have infinite number of coupling constants.
The Kaehler potential gives the Kaehler metric and Ricci tensor as
follows:
The solution of the β＝０ equation satisfies the following equation:
Here we introduce a parameter which corresponds to the anomalous
dimension of the scalar fields as follows:
When N=1, the function f(x) is given in closed form
The target manifold takes the form of a
semi-infinite cigar with radius
.
It is embedded in 3-dimensional flat
Euclidean spaces.
Witten Phys.Rev.D44 (1991) 314
This solution has been discussed in other context.
They consider the non-linear sigma model coupled with dilaton.
Witten Phys.Rev.D44 (1991) 314
Kiritsis, Kounnas and Lust Int.J.Mod.Phys.A9 (1994) 1361
Hori and Kapustin :JHEP 08 (2001) 045
In k>>1 region, we can use the perturbative renormalization method
and obtain 1-loop b function:
If one prefers to stay on a flat world-sheet, one may say that a nontrivial dilaton gradient in space-time is equivalent to assigning a
non-trivial Weyl transformation law to target space coordinates.
Our parameter a (anomalous dim.) corresponds to k as follow.
4.3-dimensional case
The 3-dimensional non-linear sigma models are nonrenormalizable
within the perturbative method. We need some nonperturbative
renormalization methods.
Similarly to 2-dimenion, we obtain the nonperturbative b function
for 3-dimensional non-linear sigma models.
When the target space is an Einstein-Kaehler manifold,
the βfunction of the coupling constant is obtained.
Einstein-Kaehler condition:
If the constant h is positive, there are two fixed points:
At UV fixed point
The value of h for hermitian symmetric spaces.
G/H
Dimensions(complex)
h
SU(N)/[SU(N-1)×U(1)]
N-1
N
SU(N)/SU(N-M)×U(M)
M(N-M)
N
SO(N)/SO(N-2)×U(1)
N-2
N-2
Sp(N)/U(N)
SO(2N)/U(N)
E6/[SO(10) ×U(1)]
E7/[E6×U(1)]
N(N+1)/2
N(N+1)/2
16
27
N+1
N-1
12
18
The CP model :SU(N+1)/[SU(N) ×U(1)]
N
If the constant h is positive, it is possible to take the continuum limit
by choosing the cutoff dependence of the bare coupling constant as
M is a finite mass scale.
We derive the action of the conformal field theory corresponding to
the fixed point of the b function.
To simplify, we assume SU(N) symmetry for Kaehler potential.
We substitute the metric and Ricci tensor given by this Kaehler
potential for following equation.
The following function satisfies b=0 for any values of parameter
A free parameter,
, is proportional to the
anomalous dimension.
If we fix the value of
, we obtain a conformal field theory.
We take the specific values of the parameter, the function takes
simple form.
●
This theory is equal to IR fixed point
of CPN model
●
This theory is equal to UV fixed
point of CPN model.
Then the parameter describes a marginal deformation from the IR
to UV fixed points of the CPN model in the theory spaces.
5. Summary
In this work, we consider the derivative interaction terms using
Wilsonian RG equation which has sharp cutoff.
The RG flows for some concrete models agree with the
perturbative or large-N results.
We construct a class of fixed point theory for 2- and 3dimensional supersymmetric NLσM.
These theory has one free parameter which corresponds to the
anomalous dimension of the scalar fields.
In the 2-dimensional case, these theory coincide with perturbative
1-loop βfunction solution for NLσM coupled with dilaton.
In the 3-dimensional case, the free parameter describes a
N
marginal deformation from the IR to UV fixed points of the CP
model in the theory spaces.