Abstract
We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual. The Wilson operators are the gravitational ones where gravity is written as a gauge theory in the first order Hilbert-Palatini formalism. We apply this method to compute the global conformal blocks and partial waves in 2d CFTs reproducing many of the known results.
Highlights
The gauge algebra sl(N, R) ⊕ sl(N, R) inside which the gravity sector in its Hilbert-Palatini formalism is embedded as an sl(2, R) ⊕ sl(2, R) sector [5, 6]
We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual
It is well known that the correlation function of a set of primary operators in a CFT can be decomposed into its partial-waves
Summary
We are interested in providing a prescription to compute partial waves of correlation functions of the dual CF Td in terms of the first order action of AdSd+1 gravity. If we are interested in finding the gauge field A for a given space (with given ea and ωab) we just have to solve this equation for g and use A = −dg g−1.3 Notice that the equation (2.3) for g has a gauge invariance It is covariant under an arbitrary local Lorentz transformation: ea → (Λ)acec, ωab → (Λ)acωcd(Λ−1)db + (Λ)acd(Λ−1)cb and g → Λg where Λ is any element of the subgroup SO(d + 1) and (Λ)ab are the matrix element of Λ in the vector representation defining an equivalence relation between g and Λg. In higher dimensions integrability will impose non-trivial constraints as F = 0 is not the equation on motion This coset element g turns out to be one of the ingredients in our prescription to compute boundary partial-waves.
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