A Note on the Chiral Anomaly in the AdS/CFT Correspondence and 1/N^2 Correction

TL;DR

This work investigates how the SU($N$) chiral anomaly in ${\cal N}=4$ SYM is encoded in AdS/CFT at finite $N$ by analyzing one-loop corrections to the five-dimensional Chern-Simons term in the $AdS_5 \times S^5$ background. It shows that gluon loops do not affect the coefficient while fermionic KK modes induce an integer shift, and that the complete KK spectrum yields the exact finite-$N$ shift $N^2 \to N^2-1$, arising from the unbalanced $k=0$ doubleton state and corresponding ghosts. This -1 correction is robust and localizes to the doubleton/ghost sector, with higher string states contributing no further corrections to the CS coefficient. The results provide a nontrivial check of AdS/CFT at finite $N$ and support the view that finite-$N$ effects are governed by KK/doubleton dynamics rather than the full string spectrum.

Abstract

According to the AdS/CFT correspondence,the d=4, N=4 SU(N) SYM is dual to the Type IIB string theory compactified on AdS_5xS^5. A mechanism was proposed previously that the chiral anomaly of the gauge theory is accounted for to the leading order in N by the Chern-Simons action in the AdS_5 SUGRA. In this paper, we consider the SUGRA\string action at one loop and determine the quantum corrections to the Chern-Simons action. While gluon loops do not modify the coefficient of the Chern-Simons action, spinor loops shift the coefficient by an integer. We find that for finite N, the quantum corrections from the complete tower of Kaluza-Klein states reproduce exactly the desired shift N^2 ---> N^2- 1 of the Chern-Simons coefficient, suggesting that this coefficient does not receive corrections from the other states of the string theory. We discuss why this is plausible.