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The large $N_c$ limit of four-point functions in N=4 super-Yang-Mills theory from anti-de Sitter Supergravity

Gordon Chalmers, Koenraad Schalm

TL;DR

The paper analyzes four-point functions in the AdS/CFT framework at large $N_c$ and strong ’t Hooft coupling, focusing on their imaginary parts obtained via unitarity cuts. By formulating Lorentzian AdS holography and deriving scalar and graviton kernels, the authors show that the imaginary parts of exchange diagrams factorize into products of two three-point boundary–bulk amplitudes, enabling explicit integral evaluations. The results for both scalar and graviton exchanges yield expressions in momentum space that are rational functions and logarithms, with the four-point functions not reproducing free-field results and revealing potential hidden symmetries in $N=4$ SYM at strong coupling. These findings provide a concrete framework to obtain complete correlators from their imaginary parts and suggest deep structural constraints in the strong-coupling regime, beyond conventional conformal constraints. The work also indicates that large-$N_c$ four-point functions at finite coupling differ from their free-field limits, as highlighted by the note added crediting subsequent confirmation.

Abstract

We compute the imaginary part of scalar four-point functions in the AdS/CFT correspondence relevant to N=4 super Yang-Mills theory. Unitarity of the AdS supergravity demands that the imaginary parts of the correlation functions factorize into products of lower-point functions. We include the exchange diagrams for scalars as well as gravitons and find explicit expressions for the imaginary parts of these correlators. In momentum space these expressions contain only rational functions and logarithms of the kinematic invariants, in such a manner that the correlator is not a free-field result. The simplicity of these results, however, indicate the possibility of additional symmetry structures in N=4 super Yang-Mills theory in the large $N_c$ limit at strong effective coupling. The complete expressions may be computed from the integral results derived here.

The large $N_c$ limit of four-point functions in N=4 super-Yang-Mills theory from anti-de Sitter Supergravity

TL;DR

The paper analyzes four-point functions in the AdS/CFT framework at large and strong ’t Hooft coupling, focusing on their imaginary parts obtained via unitarity cuts. By formulating Lorentzian AdS holography and deriving scalar and graviton kernels, the authors show that the imaginary parts of exchange diagrams factorize into products of two three-point boundary–bulk amplitudes, enabling explicit integral evaluations. The results for both scalar and graviton exchanges yield expressions in momentum space that are rational functions and logarithms, with the four-point functions not reproducing free-field results and revealing potential hidden symmetries in SYM at strong coupling. These findings provide a concrete framework to obtain complete correlators from their imaginary parts and suggest deep structural constraints in the strong-coupling regime, beyond conventional conformal constraints. The work also indicates that large- four-point functions at finite coupling differ from their free-field limits, as highlighted by the note added crediting subsequent confirmation.

Abstract

We compute the imaginary part of scalar four-point functions in the AdS/CFT correspondence relevant to N=4 super Yang-Mills theory. Unitarity of the AdS supergravity demands that the imaginary parts of the correlation functions factorize into products of lower-point functions. We include the exchange diagrams for scalars as well as gravitons and find explicit expressions for the imaginary parts of these correlators. In momentum space these expressions contain only rational functions and logarithms of the kinematic invariants, in such a manner that the correlator is not a free-field result. The simplicity of these results, however, indicate the possibility of additional symmetry structures in N=4 super Yang-Mills theory in the large limit at strong effective coupling. The complete expressions may be computed from the integral results derived here.

Paper Structure

This paper contains 11 sections, 81 equations.

Table of Contents

  1. Introduction
  2. Kernels in AdS${}_{d+1}$
  3. Scalars
  4. Gravitons in $h_{\mu 0}=0$ Gauge
  5. Unitarity and Cutting
  6. Factorization
  7. Scalar Exchange
  8. Graviton Exchange in $h_{\mu 0}=0$ Gauge
  9. Other Correlator examples
  10. Implications for $N=4$ SYM
  11. Conclusion