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

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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.