Abstract
We explore conformally coupled scalar theory in AdS6 extensively and their classical solutions by employing power expansion order by order in its self-interaction coupling λ. We describe how we get the classical solutions by diagrammatic ways which show general rules constructing the classical solutions. We study holographic correlation functions of scalar operator deformations to a certain 5-dimensional conformal field theory where the operators share the same scaling dimension ∆ = 3, from the classical solutions. We do not assume any specific form of the micro Lagrangian density of the 5-dimensional conformal field theory. For our solutions, we choose a scheme where we remove co-linear divergences of momenta along the AdS boundary directions which frequently appear in the classical solutions. This shows clearly that the holographic correlation functions are free from the co-linear divergences. It turns out that this theory provides correct conformal 2- and 3- point functions of the ∆ = 3 scalar operators as expected in previous literature. It makes sense since 2- and 3- point functions are determined by global conformal symmetry not being dependent on the details of the conformal theory. We also get 4-point function from this holographic model. In fact, it turns out that the 4-point correlation function is not conformal because it does not satisfy the special conformal Ward identity although it does dilation Ward identity and respect SO(5) rotation symmetry. However, in the co-linear limit that all the external momenta are in a same direction, the 4-point function is conformal which means that it satisfy the special conformal Ward identity. We inspect holographic n-point functions of this theory which can be obtained by employing a certain Feynman-like rule. This rule is a construction of n-point function by connecting l-point functions each other where l < n. In the co-linear limit, these n-point functions reproduce the conformal n-point functions of ∆ = 3 scalar operators in d = 5 Euclidean space addressed in arXiv:2001.05379.
TOWARDS THE PROOF OF AGT-W RELATION: TODA 3-POINT FUNCTION AND ${\mathcal W}_{1+\infty}$ ALGEBRA
