Non-perturbative renormalization group approach to supersymmetric nonlinear sigma models

1.导出了三维非线性σ模型的Wilson重整化群方程.共形不变场论被定义为该方程的不动点理论。Einstein-Kahler流形上的非线性sigma模型对应于紫外不动点.详细分析了目标流形为真实的二维时的共形场论。(1)当反常维数γ=-1/2时，目标流形为对称性增强的SU（2）球面. (2)当-1/2<γ<0时，目标流形是紧的，但具有锥奇点. (3)当γ=0时，对应于自由场理论，目标流形变得平坦。(4)当γ>0.4时，目标流形变得非紧.当γ=-1/2时，我们研究了在红外区所有的涨落在圆球周围的重整化群流，发现在红外区所有的涨落都是无关的，即在红外区目标流形的形状变成了圆形.在威尔逊重整化方法中，重整化等价于紫外不动点的存在，使得我们可以取非平凡的连续极限。为了证实我们的结果，我们还证明了这些模型在大N展开方法中也是可重正化的。

1. The Wilsonian renormalization group equation for three dimensional nonlinear sigma models was derived.2. Conformal invariant field theories are defined as the fixed point theory of that equation. Nonlinear sigma models on Einstein-Kahler manifolds correspond to ultraviolet fixed points.3. We have analyzed conformal field theories in details when the target manifold has real dimensions two.(1) When the anomalous dimension γ=-1/2, the target manifold is a round two sphere with enhanced symmetry SU (2).(2) When -1/2<γ<0, the target manifold is compact but has conical singularity.(3) When γ=0, the target manifold becomes flat corresponding to a free field theory.(4) The target manifold becomes noncompact when γ>0.4. When γ=-1/2, we have studied the renormalization group flow of the fluctuations around the round sphere, and found all fluctuations are irrelevant in the infrared region, that is, the shape of the target manifold becomes round shaped in the infrared region.5. In Wilsonian renormalization method, renormalizability is equivalent to the existence of ultraviolet fixed point such that we can take the nontrivial continuum limit. To confirm our result, we have also shown that these models are also renormalizable in the large N expansion method.