Supersymmetric three dimensional conformal sigma models ーSUSY07参加報告ー

Presentation on theme: "Supersymmetric three dimensional conformal sigma models ーSUSY07参加報告ー"— Presentation transcript:

1 Supersymmetric three dimensional conformal sigma models ーSUSY07参加報告ー
基礎物理学研究所　COE研究員 伊藤　悦子

2 -SUSY07@Karlsruhe- バーデン＝ヴュルテンベルク州 人口は約28万人 （京都市は146万人）
人口は約28万人　（京都市は146万人） シュトゥットガルト、マンハイムに続く第三の規模の都市 会議出席者：約500人　（うち日本人約30人） 1993年から毎年開かれている。 　SUSY phenomenology 　Higgs phenomenology 　Cosmology and Astrophysics 　Alternatives (large extra-dim. little Higgs) 　 Flavor physics 　 Theoretical model フランクフルト 6つのパラレルセッション カールスルーエ

3 Supersymmetry 超対称性：ボソン場とスピノール場の間の対称性 超空間：普通の座標 （ ） スピノール座標 （グラスマン数）（ ）
超空間：普通の座標　（　　　） 　　　　スピノール座標　（グラスマン数）（　　　　） グラスマン数の性質 超対称性変換：

4 カイラルスーパーフィールド 拘束条件　　　　　　　　　　　　　　　　を満たす場。 超対称な共変微分 を使うと を　　　に依らない場で書ける。 各成分の変換性： ：全微分

5 超対称性を保つラグランジアンの構成のため
→変換したとき全微分の形で書ければ不変。 の係数がラグランジアンの候補。 自由場の理論 最も簡単な相互作用のあるラグランジアン ；　　　　　　の任意の関数 ケーラーポテンシャル 非線形シグマ模型のラグランジアン

6 さらに複雑にすると、 とかける。 ケーラーポテンシャルのみ考える 相互作用の形を制限

7 Supersymmetric three dimensional conformal sigma models
Etsuko Itou (Kyoto U. YITP) hep-th/ Progress of Theoretical Physics Vol. 117 No. 6 (2007) 1139 : Collaborated with Takeshi Higashi and Kiyoshi Higashijima (Osaka U.) 2007/07/26 SUSY07, Karlsruhe

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

9 ２.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

10 Wilsonian Renormalization Group Equation
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 . Wegner-Houghton equation (sharp cutoff) Polchinski equation (smooth cutoff) Exact evolution equation ( for 1PI effective action)

11 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.

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

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) b(l) IR i, j=1 l We have only IR fixed point at l=0.

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

15 ３.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 At the point, the target mfd. is not locally flat.
It is convenient to rewrite the 2nd order diff.eq. to a set of 1st order diff.eq. Deformed sphere 　　　 　　　: Sphere　S2（CP1） 　 　　　: Deformed sphere 　　　 : Flat R2 e(r) At the point, the target mfd. is not locally flat. It has deficit angle Euler number is equal to S2

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.