1. Three dimensional conformal sigma models Collaborated with Takeshi Higashi and Kiyoshi Higashijima (Osaka U.) Etsuko Itou (Kyoto U. YITP) hep-th/0702188 To appear in Prog. Theor. Phys. 2007/08/10 QFT, Kinki University

2. Plan to talk 1.Introduction 2.Two dimensional cases 3.Renormalizability of three dimensional sigma model 4.Conformal sigma models 5.Summary We consider three dimensional nonlinear sigma models using the Wilsonian renormalization group method. In particularly, we investigate the renormalizability and the fixed point of the models.

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

4. We divide all fields 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. Wilsonian Renormalization Group Equation

5. 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, … 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

6. Canonical scaling ： Normalize kinetic terms Quantum correction Wegner-Houghton eq In this equation, all internal lines are the shell modes which have nonzero values in small regions. More than two loop diagrams vanish in the limit. This is exact equation. We can consider (perturbatively) nonrenormalizable theories.

7. 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

8. “Renormalizability” and Continuum limit ● critical line ・・・ EAB ● renormalized trajectory ● fixed point ・・・ A, B,  line The renormalization group flow which can be extrapolated back to critical surface defines a renormalized theory. Two dimensional theory space Renormalizable UV fixed point

9. 2. Two dimensional cases Non-linear sigma models with N=2 SUSY in 3D (2D) is defined by Kaehler potential. The scalar field has zero canonical dimension. In perturbative analysis, the 1-loop  function is proportional to the Ricci tensor of target spaces. The perturbative results Alvarez-Gaume, Freedman and Mukhi Ann. of Phys. 134 (1982) 392 Perturbatively renormalizable

10. Beta function from WRG K.Higashijima and E.I. Prog. Theor. Phys. 110 (2003) 107 Here we introduce a parameter which corresponds to the anomalous dimension of the scalar fields as follows: When N=1, 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 Ricci Flat solution Fixed Point Theories

11. 3.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

12. The CP N-1 model :SU(N)/[SU(N-1) ×U(1)] From this Kaehler potential, we derive the metric and Ricci tensor as follow: Beta fn. from WRG (Ricci soliton equation) Renormalization condition

13. When the target space is an Einstein-Kaehler manifold, the βfunction of the coupling constant is obtained. Einstein-Kaehler condition: The constant h is negative (example Disc with Poincare metric) We have only IR fixed point at   i, j=1 IR

14. If the constant h is positive, there are two fixed points: At UV fixed point 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. Renormalizable IR

15. 4.Conformal Non-linear sigma models Fixed point theory obtained by solving an equation At Fixed point theories have Kaehler-Einstein mfd. with the special value of the radius. Ｃ is a constant which depends on models. Hermitian symmetric space (HSS) ・・・・ A special class of Kaehler- Einstein manifold with higher symmetry

16. New fixed points (γ≠-1/2)  Two dimensional fixed point target space for The line element of target space RG equation for fixed point e(r)

17. It is convenient to rewrite the 2nd order diff.eq. to a set of 1st order diff.eq. : Sphere S 2 （ CP 1 ） : Deformed sphere : Flat R 2 e(r) Deformed sphere At the point, the target mfd. is not locally flat. It has deficit angle. Euler number is equal to S 2

18. Summary We study a perturbatively nonrenormalizable theory (3-dim. NLSM) using the WRG method. Some three dimensional nonlinear sigma models are renormalizable within a nonperturbative sense. We construct a class of 3-dim. conformal sigma models.