Wilsonian approach to Non-linear sigma models

1. 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.) 2008/2/2７, National Taiwan Normal University

2. 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

3. 1.Introduction Nonlinear Sigma model 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 (perturbatively renormalizable) 3-dim. Non-linear sigma model

4. The motivation from the point of view of non-linear 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?

5. 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 .

6. 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… local potential, Nambu-Jona-Lasinio, NLσM T.Morris, K. Itoh, Y. Igarashi, H. Sonoda, M. Bonini,… YM theory, QED, SUSY… C. Wetterich, M. Reuter, N. Tetradis, J. Pawlowski,… quantum gravity, Yang-Mills theory, higher-dimensional gauge theory…

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

8. Approximation method: Symmetry and Derivative expansion Consider a single real scalar field theory that is invariant under We expand the most generic action as Local Potential term Non-linear sigma model K.Aoki Int.J.Mod.Phys. B14 (2000) 1249 T.R.Morris Int.J.Mod.Phys. A9 (1994) 2411 Local potential approximation : In this work, we expand the action up to second order in derivative and assume it =2 supersymmetry.

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

10. 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. Chiral Super Field Kaehler terms Example Super potential terms Example WE DROP!!

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

12. 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( ) .

13. 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.

14. 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.

15. 3. Fixed points with U(N) symmetry 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

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

17. 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: For any value of a, the theory is conformal. Case 1 a>0 When N=1, the function f(x) is givenin 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

18. Case 2 a<0 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.

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

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

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

22. The b function and anomalous dimension of scalar field are given by There are two fixed points:

23. 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 b function can be rewritten

24. The value of h for hermitian symmetric spaces.

25. Because only l depends on t, the WRG eq. can be rewritten We obtain the anomalous dimension and b function of l:

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

27. The constant h is negative (example Disc with Poincare metric) i, j=1 b(l) l We have only IR fixed point at l=0. nonrenormalizable

28. 5.U(N) symmetric solution of WRG equatrion We derive the action of the conformal field theory corresponding to the fixed point of the b function.

29. To simplify, we assume U(N) symmetry for Kaehler potential. Where, The function f(x) have infinite number of coupling constants. The b=0 can be written

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

31. Similarly, we can fix all coupling constant using order by order. The following function satisfies b=0 for any values of parameter If we fix the value of , we obtain a conformal field theory.

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

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

34. “Deformed” sphere e(r) At the point, the target mfd. has conical singularity. It has deficit angle. Euler number is equal to S2

35. e(r) Non-compact manifold

36. Summary of the geometry of the conformal sigma model : SphereS2（CP1） :Deformed sphere :Flat R2 : Non-compact

37. Summary and Discussions Using the WRG, we can discuss broad class of the NLsigmaM. 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.

38. Three-dimensional nonlinear sigma models We found the NLσMs on Einstein-Kaehler manifolds with positive scalar curvature are renormalizable in three dimensions. We constructed the three dimensional conformal NLσMs. : SphereS2（CP1） :Deformed sphere :Flat R2 : Non-compact Future problems: RG flow around the novel CFT: stability, marginal operator… The properties of a 3-dimensional CFT