Drawing on analogies with the commutative case, the Wilsonian picture of renormalization is developed for noncommutative scalar field theory. The dimensionful noncommutativity parameter, theta, induces several new features. Fixed-points are replaced by `floating-points' (actions which are scale independent only up to appearances of theta written in cutoff units). Furthermore, it is found that one must use correctly normalized operators, with respect to a new scalar product, to define the right notion of relevance and irrelevance.
In this framework it is straightforward and intuitive to reproduce the classification of operators found by Grosse & Wulkenhaar, around the Gaussian floating-point. The one-loop beta-function of their model is computed directly within the exact renormalization group, reproducing the previous result that it vanishes in the self-dual theory, in the limit of large cutoff. With the link between this methodology and earlier results made, it is discussed how the vanishing of the beta-function to all loops, as found by Disertori et al., should be interpreted in a Wilsonian framework.
Wilsonian Effective Actions and the IR/UV Mixing in Noncommutative Gauge Theories
利用背景场微扰理论研究了任意物质含量的非对易规范理论的Wilson有效作用量。在单圈水平上确定了Wilson耦合常数和规范玻色子极化张量作为动量标度k的函数，并研究了它们在θ k ->0时的短程行为，其中θ是非对易性参数.非对易场论中的短程自由度和长程自由度之间的混合违反了威尔逊作用量的普适性，并导致了IR奇点。我们发现，与已知结果一致，在超对称规范理论中，二次IR发散被抵消。对数发散在质量形变N=4理论中消失，但在其他有限N=2理论中不消失。接下来我们集中讨论有限N=2和质量形变N=4的具有大质量超多重态的超对称U（1）规范理论。威尔逊运行耦合表现出一个非平凡的阈值行为，远低于noncomutativity标度，最终成为平坦的极红外在N=4理论，但不是在N=2理论。这被解释为非奇异交换极限的（不）存在性，其中该理论由交换N=2纯U（1）理论描述。我们期望我们对有限理论的分析对微扰论中的所有阶都是精确的。
We show a possible way to build the AdS/CFT correspondence starting from the quantum field theory side based on renormalization group approach. An extra dimension is naturally introduced in our scheme as the renomalization scale. The holographic wave equations are derived, with the potential term being determined by the QFT properties. We discover that only around the fixed point, i.e. the conformal limit, the potential in the bulk equations can be fully constrained, and upon this foundation, the correspondence is build. We demonstrate this fact using a 3D scalar theory in which, besides the t…
We extend the relation between stochstic quantization and Holographic Wilsonian Renormalization Group to the massive fermion case in Euclidean AdS space. The original suggestion about the explicit relation between the two theories is in arXiv:1209.2242. To give a fermion bulk action the Dirichlet boundary condition and the Neumann boundary condition, there are some additional terms to the bulk action. We have confirmed that the relation also hold for massive fermion case.
The Wilsonian renormalization group approach to matrix models is outlined and applied to multitrace matrix models with emphasis on the computation of the fixed points which could describe the phase structure of noncommutative scalar phi-four theory.
We consider a functional relation between a given Wilsonian RG flow, which has to be related to a specific coarse-graining procedure, and an infinite family of (UV cutoff) scale dependent field redefinitions. Within this framework one can define a family of Wilsonian proper-time exact RG equations associated to an arbitrary regulator function. New applications of these RG flow schemes to the Ising Universality class in three dimensions in the derivative expansion are shortly illustrated.
The gradient flow is the evolution of fields and physical quantities along a dimensionful parameter~$t$, the flow time. We give a simple argument that relates this gradient flow and the Wilsonian renormalization group (RG) flow. We then illustrate the Wilsonian RG flow on the basis of the gradient flow in two examples that possess an infrared fixed point, the 4D many-flavor gauge theory and the 3D $O(N)$ linear sigma model.
A heavy quark moving through a strongly coupled deconfined plasma has a holographic dual description as a string moving in a black brane geometry. We apply the holographic Wilsonian renormalization method to derive a holographic effective string action dual to the heavy quark. The effective action only depends on the geometry between the black brane horizon and a cutoff localized in the radial direction, corresponding to the IR of the dual theory. We derive RG flow equations for the coefficients in the effective action and show that the force acting on the heavy quark is independent of the pos…
For local conformal field theories, it is shown how to construct an expression for the energy-momentum tensor in terms of a Wilsonian effective Lagrangian. Tracelessness implies a single, unintegrated equation which enforces both the Exact Renormalization Group equation and its partner encoding invariance under special conformal transformations.
We study a mathematical relationship between holographic Wilsonian renormalization group and stochastic quantization framework. We extend the original proposal given in arXiv:1209.2242 to interacting theories. The original proposal suggests that fictitious time(or stochastic time) evolution of stochastic 2-point correlation function will be identical to the radial evolution of the double trace operator of certain classes of holographic models, which are free theories in AdS space. We study holographic gravity models with interactions in AdS space and establish a map between the holographic ren…
Holographic model of massive scalar field with its self-interaction $λϕ^n$ in AdS space is able to give a logarithmic scale dependence to marginal multi trace deformation couplings on its dual conformal field theory, where $λ$ is the self-interaction coupling of the scalar field, $ϕ$ and $n$ is an integral number. In arXiv:1501.06664, the authors realize this feature by looking at bulk scalar solutions near AdS boundary imposing a specific boundary condition between the coefficients of non-normalizable and normalizable modes of the scalar field excitations. We study the same holographic model …
In holographic duality, dynamics along the emergent extra-dimensional space describes a renormalization group (RG) flow of the corresponding quantum field theory (QFT). Following this idea, we develop an emergent holographic description of a QFT, where not only the information of the RG flow is introduced into an IR holographic dual effective field theory (HDEFT), but also the UV information of the QFT is encoded in the HDEFT through the IR boundary condition. In particular, we argue that this dual holographic construction is self-consistent within the assumption of bulk locality, showing the …
Stochastic quantization and holographic Wilsonian renormalization group of scalar theory with generic mass, self-interaction and multiple trace deformation
For conformal field theories, it is shown how the Ward identity corresponding to dilatation invariance arises in a Wilsonian setting. In so doing, several points which are opaque in textbook treatments are clarified. Exploiting the fact that the Exact Renormalization Group furnishes a representation of the conformal algebra allows dilatation invariance to be stated directly as a property of the action, despite the presence of a regulator. This obviates the need for formal statements that conformal invariance is recovered once the regulator is removed. Furthermore, the proper subset of conforma…
Elimination of IR/UV via Gravity in Noncommutative Field Theory
具有非对易几何的粒子物理模型通常会受到红外和紫外自由度的明显非威尔逊耦合的影响，称为“IR/UV问题”，这往往会损害它们的唯象相关性。在这封信中，我们明确地展示了如何通过将NCG与引力耦合来纠正这一点。在最简单的情况下，拉格朗日乘上一个非常数的背景度量;在$ϕ-4$理论中，$\int d^4 x ϕ\star ϕ= \int d^4 x ϕ^2$不再成立的定理，场传播子被一个因子修改，该因子取决于NCG和度量的变化。当传播动量逐渐变大时，该因子的适当限制将根除IR/UV问题。由于引力和NCG相互耦合，人们可能会期望在度规中出现反对称分量。宇宙学的影响，这种随后进行了讨论。
