Conformal fields from AdS flows - Construction of conformal field theories using renormalisation group flow equations in Anti-de-Sitter space

共形场理论(CFT)有着广泛的应用，包括凝聚态系统、量子色动力学和二维量子引力。由于其高度的对称性，这极大地限制了关联函数的形式，CFT也可能是第一个严格构建在四个维度上的相互作用量子场论。一个重要的例子是规范群为SU(N)的极大超对称杨-米尔斯理论，它在平面极限N→∞上表现出一种可积结构，即不受共形对称性固定的数据(共形数据)可以作为代数方程的解得到。这是一个非常活跃的研究领域，但在这个极限下，理论不是么正的，人们还需要能够计算1/N修正。这一建议的总体目的是推导一个描述保形数据随N变化的积分-微分公式，从而允许系统地计算对平面极限的1/N中所有阶的修正，甚至数值地确定有限N的保形数据(由平面极限决定初始条件)，从而完全解决了该理论。虽然有一个类似的公式来确定共形数据随耦合常数的变化，但它不能直接应用，因为该公式所需的各种非简并条件对于初始条件(由自由理论确定)不能满足。在平面极限N→∞中，更有可能满足这样的非简并条件.为了推导这个公式，我想用重整化群流方程首先在反德西特(ADS)空间(微扰理论中的所有阶)中构造一个量子场论，然后将这个理论限制在ADS的共形边界上.虽然理论的对称性通常在量子理论构建的中间步骤中被破坏，如果出现反常，最终可能无法恢复，但时空对称性通常可以始终保持。通过将量子理论从ADS限制到它的共形边界，将该理论的ADS时空对称性提高到共形对称性。因此，这种方法提供了一种新的方法来系统地构造明显无异常的CFT，并且本身就已经构成了一个重要的结果。在第二步中，我想研究ADS理论的算符乘积展开，并推导出算符乘积系数随耦合常数的变化公式。这推广了平坦空间中的相应公式，该公式也是用重整化群流方程得到的，是描述共形数据变化的公式的前身。最后，利用ADS/CFT对应关系，我将确定描述OPE系数随N的变化的双CFT的模拟公式，并由此确定保形数据的公式。

Conformal field theories (CFTs) have found a variety of applications, including condensed matter systems, quantum chromodynamics and two-dimensional quantum gravity. Due to their high amount of symmetry, which greatly constrains the form of correlation functions, a CFT may also be the first interacting quantum field theory to be rigorously constructed in four dimensions. An important example is maximally supersymmetric Yang-Mills theory with gauge group SU(N), which in the planar limit N→∞ shows an integrable structure, i.e., the data not fixed by conformal symmetry (the conformal data) can be obtained as solution of algebraic equations. This is a highly active research field, but in this limit the theory is not unitary and one also needs to be able to calculate 1/N-corrections. The overall aim of this proposal is to derive an integro-differential formula describing the change of the conformal data with N, which would then allow to systematically calculate corrections to all orders in 1/N to the planar limit, or even to determine the conformal data for finite N numerically (with the planar limit determining the initial conditions), and thus solve the theory completely. While a similar formula exists to determine changes of the conformal data with the coupling constant, it can not be applied straightforwardly since various non-degeneracy conditions, which are needed for this formula, are not fulfilled for the initial conditions (determined by the free theory). In the planar limit N→∞, it is much more likely that such non-degeneracy conditions can be fulfilled.To derive this formula, I want to use the renormalisation group flow equations to first construct a quantum field theory in Anti-de-Sitter (AdS) space (to all orders in perturbation theory), and then restrict this theory to the conformal boundary of AdS. While in general symmetries of the theory are broken in intermediate steps of the construction of the quantum theory, and may not be restored in the end if there is an anomaly, spacetime symmetries can usually be maintained throughout. By restricting the quantum theory from AdS to its conformal boundary, the AdS spacetime symmetry of the theory is enhanced to a conformal symmetry. This method therefore provides a new way to systematically construct CFTs which are manifestly non-anomalous, and would already constitute an important result in its own right. In a second step, I want to investigate the operator product expansion (OPE) for the AdS theory, and derive a formula for the changes of the OPE coefficients with the coupling constant. This generalises the corresponding formula in flat space, which was also obtained using renormalisation group flow equations, and is a precursor of the formula describing the change of the conformal data. Lastly, using the AdS/CFT correspondence I will determine the analogue formula for the dual CFT that describes changes of the OPE coefficients with N, and from this the formula for the conformal data.