The nonperturbative analyses for lower dimensional non-linear sigma models Etsuko Itou (Osaka University) 1.Introduction 2.The WRG equation for NLσM 3.Fixed points with U(N) symmetry 4.3-dimensional case 5.The other nonperturbative analysis 6. 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. Local potential term Non-linear sigma model It corresponds to next-to-leading order approximation in derivative expansion. K.Aoki Int.J.Mod.Phys. B14 (2000) 1249 T.R.Morris Int.J.Mod.Phys. A9 (1994) 2411 Local potential approximation :

● In perturbative analysis, the 1-loop  function for 2-dimensional non-linear sigma model proportional to Ricci tensor of target spaces. ⇒ Ricci Flat Is there the other fixed point? The point of view of non-linear sigma model: The perturbative results Alvarez-Gaume, Freedman and Mukhi Ann. of Phys. 134 (1982) 392 Two-dimensional case

●The 3-dimensional non-linear sigma models are nonrenormalizable within the perturbative method. We need some nonperturbative renormalization methods. Three-dimensional case 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 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  is changed to  t  exp  t . K.Aoki Int.J.Mod.Phys. B14 (2000) 1249

To obtain the WRG eq., we integrate shell mode. only Field rescaling effects to normalize kinetic terms. The Wilsonian RG equation is written as follow: 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 Where K is Kaehler potential and  is chiral superfield. i=1 ～ N ： N is the dimensions of target spaces

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  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 We derive the action of the conformal field theory corresponding to the fixed point of the  function. To simplify, we assume U(N) symmetry for Kaehler potential. where The perturbative βfunction follows the Ricci-flat target manifolds. Ricci-flat

The Kaehler potential gives the Kaehler metric and Ricci tensor as follows: The function f(x) have infinite number of coupling constants.

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

4.3-dimensional case Similarly to 2-dimenion, we obtain the nonperturbative  function for 3-dimensional non-linear sigma models. The 3-dimensional non-linear sigma models are nonrenormalizable within the perturbative method. We need some nonperturbative renormalization methods.

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

G/HDimensions(complex)h SU(N)/[SU(N-1)×U(1)] N-1N SU(N)/SU(N-M)×U(M) M(N-M)N SO(N)/SO(N-2)×U(1)N-2 Sp(N)/U(N)N(N+1)/2N+1 SO(2N)/U(N)N(N+1)/2N-1 E 6 /[SO(10) ×U(1)]1612 E 7 /[E 6 ×U(1)]2718 The value of h for hermitian symmetric spaces.

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. The CP N model :SU(N+1)/[SU(N) ×U(1)]

To simplify, we assume SU(N) symmetry for Kaehler potential. We derive the action of the conformal field theory corresponding to the fixed point of the  function. We substitute the metric and Ricci tensor given by this Kaehler potential for following equation.

The following function satisfies  =0 for any values of parameter If we fix the value of, we obtain a conformal field theory. A free parameter,, is proportional to the anomalous dimension.

We take the specific values of the parameter, the function takes simple form. This theory is equal to IR fixed point of CP N model ● ● This theory is equal to UV fixed point of CP N model. Then the parameter describes a marginal deformation from the IR to UV fixed points of the CP N model in the theory spaces.

5.The other nonperturbative analysis At UV fixed point To check the existence of the UV fixed point, we consider the model using large-N expansion.

CP N-1 model U(1) gauge auxiliary field: Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （ 2000 ） 1283 The cases of Using the component fields

This model also has two phases: :SU(N)-symmetric, massive phase :SU(N) broken, massless phase We can assign U(1) charge for chiral and anti- chiral superfields as follow. U(1) symmetric phase U(1) broken phase

The gap eq. The β function of this model has no next-to-leading corrections. Symmetric and massive phase Broken and massless phase Is supersymmetry responsible for the vanishing of the next-to-leading order corrections to βfunction of the model?

model ´t Hooft coupling ： The examples of the Einstein-Kaehler cases 1/N next-to-leading? WRG result

Q(N-2) model model O(N) condition: There are two multiplier superfields.

There are three phases. SO(N) broken, massless theory New phase SO(N)-symmetric,massive theory ① ② The differences between ① and ② phases: ① In effective action, the gauge fields have Chern-Simons interaction. The Parity is broken. ② The gauged U(1) symmetry is broken. Symmetric and massive phase Broken and massless phase （①＆②）

We calculate the βfunction to next-to-leading order and obtain the next-to-leading order correction. Some 3-dimesional supersymmetric sigma models have the next-to-leading order correction of the βfunction. Next-to-leading order corrections to the propagator.

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 3- dimensional 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 marginal deformation from the IR to UV fixed points of the CP N model in the theory spaces.

We found the NLσMs on Einstein-Kaehler manifolds with positive scalar curvature are renormalizable in three dimensions. We also discussed two models by using the other nonperturbative method, large-N expansion. We investigated the supersymmetric nonlinear sigma model and found that the WRG analysis is very powerful to reveal various nonperturbative aspects of field theories in two- and three-dimensions.