Supersymmetric three dimensional conformal sigma

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
-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）
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）
フランクフルト
1993年から毎年開かれている。
SUSY phenomenology
Higgs phenomenology
6つのパラレル Cosmology and Astrophysics
セッション
Alternatives (large extra-dim. little Higgs)
カールスルーエ
Flavor physics
Theoretical model
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）（
グラスマン数の性質
超対称性変換：
）
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
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
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
Supersymmetric three
dimensional conformal sigma
models
Etsuko Itou (Kyoto U. YITP)
hep-th/0702188
Progress of Theoretical Physics
Vol. 117 No. 6 (2007) 1139 :
Collaborated with
Takeshi Higashi and Kiyoshi Higashijima
(Osaka U.)
2007/07/26 SUSY07, Karlsruhe
1.Introduction
Non-Linear Sigma Model
Bosonic Non-linear sigma model
The target space ・・・O(N) model
2-dim. Non-linear sigma model
(perturbatively
renormalizable)
Toy model of 4-dim. Gauge theory
(Asymptotically free, instanton, mass gap etc.)
Polyakov action of string theory
２.Three dimensional cases
(renormalizability)
The scalar field has nonzero canonical dimension.
We need some nonperturbative renormalization
methods.
WRG approach
Our works
Large-N expansion
Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （2000）1283
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)
Wegner-Houghton eq
Quantum correction
Canonical scaling： Normalize kinetic terms
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.
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:
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.
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.
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
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)
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
e(r)
: Flat R2
At the point, the target mfd. is not locally flat.
It has deficit angle.
Euler number is equal to S2
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.

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