论文标题
扰动4D共形场理论和代数的表示理论
Perturbative 4D conformal field theories and representation theory of diagram algebras
论文作者
论文摘要
通过二维$ so（4,2）$ ecorivariant拓扑场理论（TFT2）中的振幅显示，自由四维形成型场理论（CFT4）的相关因子已证明是通过使用顶点操作员形式形式的相关器来给出的。我们表明，使用两个表示理论构建体可以将其扩展到扰动性相互作用的共形场理论。在非添加异常的尺寸的情况下，共形代数的偶发性变形可容纳复合算子的均值结构。在$ \ Mathcal {n} = 4 $ sym和Wilson-Fischer固定点附近的$ \ Mathcal {n} = 4 $ sym的扇区内给出了脱层变形的显式表达式。 The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra $ {\bf U}_{*} $, abstracted from $ Uso(d)$ in the limit of large integer $d$, which admits extension of $ Uso(d)$ representation theory to general real (or complex) $d$.代数通过振荡器实现与$（d）$ ecurivariant地图和Brauer类别图相关。在图表代数作用于基本图表示的张量产物的情况下，构建了张量表示。定义了类似的图解代数$ {\ bf u} _ {\ star，2} $，与$ uso的一般$ d $扩展相关（d，2）$，并描述了与Wilson-Fischer固定点相关的一些最低权重表示。
The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional $so(4,2)$ equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of $ \mathcal{N} =4 $ SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra $ {\bf U}_{*} $, abstracted from $ Uso(d)$ in the limit of large integer $d$, which admits extension of $ Uso(d)$ representation theory to general real (or complex) $d$. The algebra is related, via oscillator realisations, to $so(d)$ equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra ${\bf U}_{\star ,2}$, related to a general $d$ extension for $ Uso(d,2)$ is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.
