Thermal correlation functions in CFT and factorization

Abstract

We study 2-point and 3-point functions in CFT at finite temperature for large dimension operators using holography. The 2-point function leads to a universal formula for the holographic free energy in d dimensions in terms of the c-anomaly coefficient. By including α corrections to the black brane background, one can reproduce the leading correction at strong coupling. In turn, 3-point functions have a very intricate structure, exhibiting a number of interesting properties. In simple cases, we find an analytic formula, which reduces to the expected expressions in different limits. When the dimensions satisfy ∆ i = ∆ j + ∆ k , the thermal 3-point function satisfies a factorization property. We argue that in d > 2 factorization is a reflection of the semiclassical regime.

Holographic thermal correlators revisited

“…For the leading term, the euclidean version of the correlation function is in terms of real geodesics which live in the cigar geometry. 6 As shown in [2,3], these precisely reproduce the expectations from CFT not only showing the expected structure in terms of Gegenbauer polynomials but also recovering the correct numerical values of the central charges. In turn, for the subleading term in fig.…”

“…Very recently thermal 2-point functions have been studied for holographic CFT's in [2,3]. 3 In principle, these are read-off from the solution to the corresponding wave equation with the usual holographic recipe.…”

“…Very recently thermal 2-point functions have been studied for holographic CFT's in [2,3]. 3 In principle, these are read-off from the solution to the corresponding wave equation with the usual holographic recipe. As this is a daunting task (specially in order to explore the whole thermal circle), [2,3] specialized to 2-point functions of scalar operators of large conformal dimension (but still much smaller than the central charge of the theory).…”

“…3 In principle, these are read-off from the solution to the corresponding wave equation with the usual holographic recipe. As this is a daunting task (specially in order to explore the whole thermal circle), [2,3] specialized to 2-point functions of scalar operators of large conformal dimension (but still much smaller than the central charge of the theory). In that limit, the Klein-Gordon equation can be solved through the WKB approximation, which in the end becomes the (exponentiated) geodesic length.…”

“…(1) following [2,3], including a detailed discussion of the vanishing of that diagram for off-diagonal correlators. In section 4 we discuss the computation of the subleading term in fig.…”