From Free Fields to $AdS$

Free ${\cal N}=4$ Super Yang-Mills theory (in the large $N$ limit) is dual to an, as yet, intractable closed string theory on $AdS_5\times S^5$. We aim to implement open-closed string duality in this system and thereby recast the free field correlation functions as amplitudes in $AdS$. The basic strategy is to implement this duality directly on planar field theory correlation functions in the worldline (or first quantised) formulation. The worldline loops (remnants of the worldsheet holes) close to form tree diagrams. These tree diagrams are then to be manifested as tree amplitudes in $AdS$ by a change of variables on the worldline moduli space (i.e. Schwinger parameter space). Restricting to twist two operators, we are able to carry through this program for two and three point functions. However, it appears that this strategy can be implemented for four and higher point functions as well. An analogy to electrical networks is very useful in this regard.