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Extremal Correlators in the AdS/CFT Correspondence

E. D'Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, L. Rastelli

TL;DR

This paper resolves a central puzzle in AdS/CFT: extremal correlators of chiral primaries are finite and governed by single-trace operators despite naive divergences and potential multi-trace mixing. By carefully regularizing the AdS integrals and analyzing boundary contributions, the authors show extremal 3-point functions arise from single-trace exchanges, with non-extremal cases dominated by bulk cubic couplings and extremal cases by regulated boundary terms. They extend these insights to extremal n-point functions, arguing a factorized space-time form with a non-renormalized coefficient, a prediction supported by OPE-based non-renormalization arguments. The results reinforce the picture that extremal correlators provide a sharp probe of operator mixing and yield robust, coupling-independent predictions in the strong-coupling regime of ${ m{ ext{SYM}}}$ at large $N$.

Abstract

The non-renormalization of the 3-point functions $tr X^{k_1} tr X^{k_2} tr X^{k_3}$ of chiral primary operators in N=4 super-Yang-Mills theory is one of the most striking facts to emerge from the AdS/CFT correspondence. A two-fold puzzle appears in the extremal case, e.g. k_1 = k_2 + k_3. First, the supergravity calculation involves analytic continuation in the k_i variables to define the product of a vanishing bulk coupling and an infinite integral over AdS. Second, extremal correlators are uniquely sensitive to mixing of the single-trace operators $tr X^k$ with protected multi-trace operators in the same representation of SU(4). We show that the calculation of extremal correlators from supergravity is subject to the same subtlety of regularization known for the 2-point functions, and we present a careful method which justifies the analytic continuation and shows that supergravity fields couple to single traces without admixture. We also study extremal n-point functions of chiral primary operators, and argue that Type IIB supergravity requires that their space-time form is a product of n-1 two-point functions (as in the free field approximation) multiplied by a non-renormalized coefficient. This non-renormalization property of extremal n-point functions is a new prediction of the AdS/CFT correspondence. As a byproduct of this work we obtain the cubic couplings $t φφ$ and $s φφ$ of fields in the dilaton and 5-sphere graviton towers of Type IIB supergravity on $AdS_5 \times S^5$.

Extremal Correlators in the AdS/CFT Correspondence

TL;DR

This paper resolves a central puzzle in AdS/CFT: extremal correlators of chiral primaries are finite and governed by single-trace operators despite naive divergences and potential multi-trace mixing. By carefully regularizing the AdS integrals and analyzing boundary contributions, the authors show extremal 3-point functions arise from single-trace exchanges, with non-extremal cases dominated by bulk cubic couplings and extremal cases by regulated boundary terms. They extend these insights to extremal n-point functions, arguing a factorized space-time form with a non-renormalized coefficient, a prediction supported by OPE-based non-renormalization arguments. The results reinforce the picture that extremal correlators provide a sharp probe of operator mixing and yield robust, coupling-independent predictions in the strong-coupling regime of at large .

Abstract

The non-renormalization of the 3-point functions of chiral primary operators in N=4 super-Yang-Mills theory is one of the most striking facts to emerge from the AdS/CFT correspondence. A two-fold puzzle appears in the extremal case, e.g. k_1 = k_2 + k_3. First, the supergravity calculation involves analytic continuation in the k_i variables to define the product of a vanishing bulk coupling and an infinite integral over AdS. Second, extremal correlators are uniquely sensitive to mixing of the single-trace operators with protected multi-trace operators in the same representation of SU(4). We show that the calculation of extremal correlators from supergravity is subject to the same subtlety of regularization known for the 2-point functions, and we present a careful method which justifies the analytic continuation and shows that supergravity fields couple to single traces without admixture. We also study extremal n-point functions of chiral primary operators, and argue that Type IIB supergravity requires that their space-time form is a product of n-1 two-point functions (as in the free field approximation) multiplied by a non-renormalized coefficient. This non-renormalization property of extremal n-point functions is a new prediction of the AdS/CFT correspondence. As a byproduct of this work we obtain the cubic couplings and of fields in the dilaton and 5-sphere graviton towers of Type IIB supergravity on .

Paper Structure

This paper contains 15 sections, 72 equations.

Table of Contents

  1. Introduction
  2. Multi--trace BPS operators and Mixing
  3. Large $N$ counting
  4. Supergravity computation of $\langle {\cal O}_{t,s}^{k_1} {\cal O}_\phi^{k_2} {\cal O}_\phi^{k_3} \rangle$
  5. Set--up
  6. Constraints
  7. Cubic action
  8. Dimensional reduction
  9. Evaluation of the action
  10. Computing the 3--point function
  11. The three--point function for $k_1<k_2+k_3$
  12. The extremal case $k_1=k_2+k_3$
  13. Discussion
  14. Non--renormalization conjectures for extremal n--point functions
  15. Appendix