Wilsonian approach to Non-linear sigma models Etsuko Itou (YITP, Japan) Progress of Theoretical Physics 109 (2003) 751 Progress of Theoretical Physics 110 (2003) 563 Progress of Theoretical Physics 117 (2007)1139 Collaborated with K.Higashijima and T.Higashi (Osaka U.) 2008/2/2 ７, National Taiwan Normal University

Plan to talk Introduction (WRG, Approximation) WRG eq. for SUSY NLsigmaM Fixed points of 2-dim. NLsigmaM Renormalizability of 3-dim. NLsigmaM Fixed points of 3-dim. NLsigmaM Summary

Bosonic Non-linear sigma model EX) The target space ・・・ O(N) model Toy model of 4-dim. Gauge theory (Asymptotically free, instanton, mass gap etc.) Polyakov action of string theory Perturbatively non-renormalizable theory 2-dim. Non-linear sigma model 1.Introduction 3-dim. Non-linear sigma model (perturbatively renormalizable) Nonlinear Sigma model

In two-dimensions, the 1-loop beta- function for 2-dimensional non-linear sigma model is proportional to Ricci tensor of target spaces. ⇒ Ricci Flat (Calabi-Yau manifold) The other fixed point? In three-dimensions, the sigma models are perturbatively nonrenormalizable. Nonperturbatively? The motivation from the point of view of non-linear sigma model

Wilsonian Renormalization Group K.Aoki Int.J.Mod.Phys. B14 (2000) 1249 We divide all fields Φ into two groups, high frequency modes and low frequency modes. The high frequency mode is integrated out.  Infinitesimal change of cutoff The partition function does not depend on Λ.

There are some Wilsonian renormalization group equations. Wegner-Houghton equation (sharp cutoff) Polchinski equation (smooth cutoff) Exact evolution equation ( for 1PI effective action) K-I. Aoki, H. Terao, K.Higashijima… T.Morris, K. Itoh, Y. Igarashi, H. Sonoda, M. Bonini,… C. Wetterich, M. Reuter, N. Tetradis, J. Pawlowski,… quantum gravity, Yang-Mills theory, higher-dimensional gauge theory… YM theory, QED, SUSY… local potential, Nambu-Jona-Lasinio, NLσM

Field rescaling effects to normalize kinetic terms. Wegner-Houghton equation Free parameter Shell modes Exact equation Wilsonian effective action has infinite number of interaction terms

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. K.Aoki Int.J.Mod.Phys. B14 (2000) 1249 T.R.Morris Int.J.Mod.Phys. A9 (1994) 2411 Local potential approximation : Local Potential term Non-linear sigma model

The motivation from the point view of Wilsonian RG Local potential term Non-linear sigma model We consider the Wilsonian effective action which has derivative interactions. K-I Aoki, T.Morris…

D=2 N =2 supersymmetric theory The number of supercharges are 4. 2-dim. N=2 SUSY theories are given from the dimensional reduction of 4-dim. N=1 SUSY theory. WE DROP!! Chiral Super Field Example Kaehler terms Super potential terms

i=1 ～ N ： N is the dimensions of target spaces Considering only Kaehler potential term corresponds to second order to derivative for scalar field. There is not local potential term. 4-dim. N=1 SUSY the target space is Kaehler manifold (one of complex mfd.). 2-dim. N=2 SUSY = For arbitrary fn. K[ ]

2.WRG equation for SUSY 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 terms cancel each other even if we do renormalization group transf.

Finally, we obtain the WRG eq. for the non-linear sigma models as follow: The beta-function for the Kaehler metric is Perturbative beta-fn.

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 Kaehler potential automatically gives the Kaehler metric and Ricci tensor. The function f(x) have infinite number of coupling constants. Where,

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 Case 1 a>0 For any value of a, the theory is conformal.

The metric and scalar curvature read The target space is embedded in 3-dim. Minkowski spaces. The vertical axis has negative signature. In the asymptotic region, the surface approach the lightcone. Although the volume integral is divergent, the distance to the boundary is finite. Case 2 a<0

Summary of the two-dimensional fixed point theory Ricci flat solution Free theory Witten’s Euclidean black hole solution Minkowskian new solution

4.Three dimensional cases (renormalizability) The scalar field has nonzero canonical dimension. We need some nonperturbative renormalization methods. WRG approach Large-N expansion Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （ 2000 ） 1283 Our works

In 3-dimension, the beta-function for Kaehler metric is written: The CP N model :SU(N+1)/[SU(N) ×U(1)] Renormalization Group Flow From this Kaehler potential, we derive the metric and Ricci tensor as follow:

There are two fixed points: The  function and anomalous dimension of scalar field are given by

Einstein-Kaehler manifolds The Einstein-Kaehler manifolds satisfy the condition If h is positive, the manifold is compact. Using these metric and Ricci tensor, the  function can be rewritten

G/HDimensions(complex)h SU(N+1)/[SU(N)×U(1)] NN+1 SU(N)/SU(N-M)×U(M) M(N-M)N SO(N+2)/SO(N)×U(1)NN 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.

Because only depends on t, the WRG eq. can be rewritten We obtain the anomalous dimension and  function of 

The constant h is positive (compact E-K)Renormalizable 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. At UV fixed point

The constant h is negative (example Disc with Poincare metric) We have only IR fixed point at   i, j=1 nonrenormalizable

5.U(N) symmetric solution of WRG equatrion 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 function f(x) have infinite number of coupling constants. The  can be written

We choose the coupling constants and anomalous dimension, which satisfy this equation.

The following function satisfies  =0 for any values of parameter Similarly, we can fix all coupling constant using order by order. 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 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.

 Two dimensional fixed point target space for The line element of target space RG equation for fixed point e(r) The target manifolds of the conformal sigma models

e(r) At the point, the target mfd. has conical singularity. It has deficit angle. Euler number is equal to S 2 “Deformed” sphere

e(r) Non-compact manifold

: Sphere S 2 （ CP 1 ） : Deformed sphere : Flat R 2 : Non-compact Summary of the geometry of the conformal sigma model

Summary and Discussions Two-dimensional nonlinear sigma models We construct a class of fixed point theory for 2-dimensional supersymmetric NLsigmaM which vanishes the nonperturbative beta- function. 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 beta-function solution for NLsigmaM coupled with dilaton. Using the WRG, we can discuss broad class of the NLsigmaM.

RG flow around the novel CFT: stability, marginal operator… The properties of a 3-dimensional CFT : Sphere S 2 （ CP 1 ） : Deformed sphere : Flat R 2 : Non-compact We constructed the three dimensional conformal NLσMs. Future problems: We found the NLσMs on Einstein-Kaehler manifolds with positive scalar curvature are renormalizable in three dimensions. Three-dimensional nonlinear sigma models