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Next-next-to-extremal Four Point Functions of N=4 1/2 BPS Operators in the AdS/CFT Correspondence

Linda I. Uruchurtu

TL;DR

This work computes next-next-to-extremal four-point functions of $1/2$-BPS operators in $ ext{N}=4$ SYM at large $N$ and strong coupling via AdS$_5 imes S^5$ supergravity. It shows the on-shell five-dimensional action has no four-derivative couplings and that the four-point amplitude splits into a free part and a dynamical part governed by a single $ar{D}$-function, in agreement with the Dolan–Nirschl–Osborn conjecture. The free-field result provides a baseline, while the holographic calculation confirms the conjectured dynamical structure and highlights cancellations between contributions, with implications for the OPE and the spectrum of long multiplets at strong coupling. The analysis also discusses potential generalizations to arbitrary weight configurations and connections to semiclassical string results, suggesting a unified strong-coupling picture for these correlators.

Abstract

Four point functions of general N=4 1/2-BPS primary fields, satisfying the next-next-to-extremality condition Δ_{1}+Δ_{2}+Δ_{3}-Δ_{4}=4 are studied at large N and strong coupling. We apply new techniques to evaluate the effective couplings in supergravity, and confirm that the four derivative couplings arising in the five-dimensional supergravity vanish on-shell. We then show that the four point amplitude resulting from supergravity naturally splits into a "free" and an interactive part which resembles an effective quartic interaction. The precise structure agrees with superconformal symmetry and supports the conjecture formulated by Dolan, Osborn and Nirschl regarding the strongly coupled form of four point correlators of chiral primary operators. We also evaluate the amplitude in large N free field SYM theory and discuss the results in the context of the correspondence.

Next-next-to-extremal Four Point Functions of N=4 1/2 BPS Operators in the AdS/CFT Correspondence

TL;DR

This work computes next-next-to-extremal four-point functions of -BPS operators in SYM at large and strong coupling via AdS supergravity. It shows the on-shell five-dimensional action has no four-derivative couplings and that the four-point amplitude splits into a free part and a dynamical part governed by a single -function, in agreement with the Dolan–Nirschl–Osborn conjecture. The free-field result provides a baseline, while the holographic calculation confirms the conjectured dynamical structure and highlights cancellations between contributions, with implications for the OPE and the spectrum of long multiplets at strong coupling. The analysis also discusses potential generalizations to arbitrary weight configurations and connections to semiclassical string results, suggesting a unified strong-coupling picture for these correlators.

Abstract

Four point functions of general N=4 1/2-BPS primary fields, satisfying the next-next-to-extremality condition Δ_{1}+Δ_{2}+Δ_{3}-Δ_{4}=4 are studied at large N and strong coupling. We apply new techniques to evaluate the effective couplings in supergravity, and confirm that the four derivative couplings arising in the five-dimensional supergravity vanish on-shell. We then show that the four point amplitude resulting from supergravity naturally splits into a "free" and an interactive part which resembles an effective quartic interaction. The precise structure agrees with superconformal symmetry and supports the conjecture formulated by Dolan, Osborn and Nirschl regarding the strongly coupled form of four point correlators of chiral primary operators. We also evaluate the amplitude in large N free field SYM theory and discuss the results in the context of the correspondence.

Paper Structure

This paper contains 19 sections, 185 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Structure of the Correlation Function
  3. Free Field Theory at large $N$
  4. Supergravity Calculation
  5. On-Shell Lagrangian
  6. Results for Exchange Integrals
  7. Contact Interactions
  8. Results for the Four Point Function
  9. Verifying the CFT predictions
  10. Discussion
  11. Harmonic Variables and Propagator Basis
  12. Integration over $S^5$
  13. Properties of $D$-Functions
  14. Various Manipulations involving $\bar{D}$-functions
  15. Quartic Interactions
  16. ...and 4 more sections

Figures (6)

  • Figure 1: Diagramatic representation for the free contribution to the process $\langle \mathcal{O}_{k+2}\mathcal{O}_{k+2}\mathcal{O}_{n-k}\mathcal{O}_{n+k}\rangle$. The graphs are arranged in four equivalence classes.
  • Figure 2: In the limit in which $k\rightarrow 0$, this diagram becomes the disconnected contribution to the four point function.
  • Figure 3: Second diagram
  • Figure 4: Witten Diagrams for the $s$-channel process. (a) exchange by a scalar with $m^2=(2k+2)(2k-2)$(b) exchange by a massive vector of mass $m_{2k+1}^2=4k(k+1)$(c) exchange by a tensor field of mass $f(2k)=4k(k+2)$
  • Figure 5: Witten Diagrams for the $t$-channel process. (a) exchange by a scalar of mass $m^2=n(n-4)$(b) exchange by a vector of mass $m_{n-1}^2=n(n-2)$(c) exchange by a tensor field of mass $f(n-2)=(n-2)(n+2)$(d) Contact diagram.
  • ...and 1 more figures