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Field Theory Tests for Correlators in the AdS/CFT Correspondence

Eric D'Hoker, Daniel Z. Freedman, Witold Skiba

TL;DR

This work investigates whether radiative corrections modify 2- and 3-point correlators of chiral primary operators $\mathrm{Tr}(X^k)$ in $\mathcal{N}=4$ SYM within the AdS/CFT framework. Using an $\mathcal{N}=1$ decomposition to manage flavor and color combinatorics, the authors compute ${\cal O}(g^2)$ corrections and demonstrate their complete cancellation for all $N$ and $k$, including correlators involving $k$-fold traces and the case with $\mathrm{Tr}(X^2)$ descendants. The results corroborate the large-$N$ fixed-$\lambda$ supergravity predictions of LMRS and extend nonrenormalization to descendents, hinting at deeper symmetries in $\mathcal{N}=4$ SYM. They also discuss limitations at higher orders, and argue that S-duality constrains nonperturbative corrections in a way that is consistent with these findings.

Abstract

The order g^2 radiative corrections to all 2- and 3-point correlators of the composite primary operators Tr X^k are computed in {\cal N} = 4 supersymmetric Yang-Mills theory with gauge group SU(N). Corrections are found to vanish for all N. For k=2 this is a consequence of known superconformal nonrenormalization theorems, and for general k the result confirms an N-to-infinity, fixed large g^2N supergravity calculation and further conjectures in hep-th/9806074. A 3-point correlator involving {\cal N} = 4 descendents of Tr X^2 is calculated, and its order g^2 contribution also vanishes, giving evidence for the absence of radiative corrections in correlators of descendent operators.

Field Theory Tests for Correlators in the AdS/CFT Correspondence

TL;DR

This work investigates whether radiative corrections modify 2- and 3-point correlators of chiral primary operators in SYM within the AdS/CFT framework. Using an decomposition to manage flavor and color combinatorics, the authors compute corrections and demonstrate their complete cancellation for all and , including correlators involving -fold traces and the case with descendants. The results corroborate the large- fixed- supergravity predictions of LMRS and extend nonrenormalization to descendents, hinting at deeper symmetries in SYM. They also discuss limitations at higher orders, and argue that S-duality constrains nonperturbative corrections in a way that is consistent with these findings.

Abstract

The order g^2 radiative corrections to all 2- and 3-point correlators of the composite primary operators Tr X^k are computed in {\cal N} = 4 supersymmetric Yang-Mills theory with gauge group SU(N). Corrections are found to vanish for all N. For k=2 this is a consequence of known superconformal nonrenormalization theorems, and for general k the result confirms an N-to-infinity, fixed large g^2N supergravity calculation and further conjectures in hep-th/9806074. A 3-point correlator involving {\cal N} = 4 descendents of Tr X^2 is calculated, and its order g^2 contribution also vanishes, giving evidence for the absence of radiative corrections in correlators of descendent operators.

Paper Structure

This paper contains 6 sections, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Correlators $\langle \mathop{\rm Tr} X^k\, \mathop{\rm Tr} X^k \rangle$
  3. The Correlators $\langle \mathop{\rm Tr} X^k \, \mathop{\rm Tr} X^k \, \mathop{\rm Tr} X^2 \rangle$
  4. The Correlators $\langle \mathop{\rm Tr} X^{k_1} \, \mathop{\rm Tr} X^{k_2} \, \mathop{\rm Tr} X^{k_3} \rangle$
  5. A descendent correlator
  6. Conclusions

Figures (4)

  • Figure 1: Born graph for the $\langle \mathop{\rm Tr}(z^1)^k\, \mathop{\rm Tr} (\bar{z}^1)^k \rangle$ correlator.
  • Figure 2: Order $g^2$ corrections to the 2-point function.
  • Figure 3: Order $g^2$ corrections to the 3-point function. In addition to gauge boson exchange diagrams (b), (c), and (d) there are analogous diagrams with quartic interactions.
  • Figure 4: Two graphs contributing to the $\langle S^{44} S^{11} \mathop{\rm Tr} (\bar{z}^1)^2 \rangle$ correlator. Solid lines represent the boson propagators, broken ones fermion propagators.