[Paper Review] A Basis of Analytic Functionals for CFTs in General Dimension

Dalimil Mazáč, Leonardo Rastelli+1 more|arXiv (Cornell University)|Oct 28, 2019

Black Holes and Theoretical Physics67 references33 citations

TL;DR

The authors construct analytic functionals forming a basis for the crossing equation in CFTs in general dimension, using a double-trace basis and two independent dual construction methods, with a link to conformal dispersion relations and Polyakov-Regge blocks.

ABSTRACT

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

Motivation & Objective

• Develop an analytic framework for the four-point crossing equation in CFTs in arbitrary spacetime dimension.
• Identify a convenient basis for the space of two-variable analytic functions with boundedness at infinity.
• Construct dual functionals to extract OPE data and derive sum rules, especially for holographic CFTs.
• Introduce Polyakov-Regge blocks and relate them to Witten diagrams to streamline functional actions.

Proposed method

• Define the function space of four-point correlators with s- and t-channel expansions and a boundedness condition at infinity.
• Propose a primal basis consisting of s- and t-channel double-trace conformal blocks and their Delta-derivatives with dimensions Delta_n = 2 Delta_phi + 2 n + ell.
• Construct the dual basis functionals by two independent algorithms: (i) kernel/contour integral representation with prescribed zeros, (ii) Polyakov-Regge block decomposition tied to dispersion relations.
• Introduce Polyakov-Regge blocks P^s and P^t that reproduce the correct double-discontinuities and relate them to exchange Witten diagrams plus Regge improvements.
• Discuss the relation to Carmi-Caron-Huot’s conformal dispersion relation and how it provides a complementary route to obtaining the dual basis.

Experimental results

Research questions

• RQ1 What is a natural, complete basis for analytic functionals acting on four-point CFT correlators in general dimension?
• RQ2 How can one explicitly construct dual functionals that isolate or suppress specific OPE data, particularly double-trace versus single-trace contributions?
• RQ3 How do Polyakov-Regge blocks encode the action of functionals on conformal blocks, and what is their relation to Witten diagrams?
• RQ4 What is the connection between the constructed functionals and the conformal dispersion relation framework?
• RQ5 Can the formalism produce concrete, model-independent sum rules that are especially useful for holographic CFTs?

Key findings

• A basis of functionals dual to the double-trace conformal blocks and their Delta-derivatives is constructed for general dimension.
• Two independent algorithms are developed to build the dual basis: (a) kernel/contour integral method with prescribed zeros, and (b) Polyakov-Regge block method linked to dispersion relations.
• Polyakov-Regge blocks are defined and shown to decompose the action of functionals on both s- and t-channel blocks, and they can be related to exchange Witten diagrams with Regge improvements.
• The approach yields exact sum rules upon applying the dual functionals to the crossing equation, with special relevance to holographic CFTs where double-trace contributions are suppressed in the dual basis.
• The work connects to the conformal dispersion relation framework of Carmi and Caron-Huot, offering a complementary route to obtain the dual basis.