Holographic correlators and their (hidden) symmetries

Abstract

In this thesis we discuss several aspects of correlators at strong coupling in three different theories: $\mathcal{N}=4$ Super Yang-Mills (SYM) in four dimensions, the D3-D7 system and the D1-D5 CFT.
In the first part we focus on $\mathcal{N}=4$ SYM. After reviewing some background material, we outline a procedure to compute tree-level $\alpha'$ corrections to the four-point function of half-BPS operators, dual - via AdS/CFT - to the scattering of four Kaluza-Klein modes in $AdS_5\times S^5$. The results enjoy remarkable features, ultimately ascribed to a hidden $10d$ conformal structure governing the dynamics of $\mathcal{N}=4$ SYM in the supergravity regime.
The method, that relies on the understanding of certain patterns in the CFT data at large $N$, also fixes uniquely a particular class of anomalous dimensions associated to long double-trace operators exchanged.
These anomalous dimensions are the zeros of a characteristic polynomial which enjoys a lot of very intriguing properties which we describe in detail.
In the second part of the thesis we explore two other conformal field theories: the D3-D7 system, described by a certain $4d$ $\mathcal{N}=2$ superconformal field theory and the D1-D5 CFT, which is a $2d$ $\mathcal{N}=(4,4)$ superconformal field theory. We show that the extreme simplicity of the tree-level dynamics in these theories - due to $8d$ and $6d$ hidden conformal symmetries - allows to define generalised Mellin transforms which make manifest many properties of the associated AdS amplitudes, such as the large $p$ limit, and - in the case of the D3-D7 CFT - Bern-Carrasco-Johannson (BCJ) and double-copy relations.
We then compute the anomalous dimensions of the long-double trace operators exchanged, and show that, like in $\mathcal{N}=4$ SYM, they are simple rational function of their quantum numbers and manifest a residual degeneracy, as a consequence of the hidden symmetries.
Lastly, we show that many of the formulae we recalled/derived in this thesis can be nicely assembled into compact expressions which interpolate between different theories.

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