A Note on the Chiral Anomaly in the AdS/CFT Correspondence and 1/N² Correction

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.