SL ( 2 , Z ) multiplets in N = 4 SYM theory

We discuss the action of SL(2, Z) on local operators in D = 4, N = 4 SYM theory in the superconformal phase. The modular property of the operator’s scaling dimension determines whether the operator transforms as a singlet, or covariantly, as part of a finite or infinite dimensional multiplet under the SL(2, Z) action. As an example, we argue that operators in the Konishi multiplet transform as part of a (p, q) PSL(2, Z) multiplet. We also comment on the non-perturbative local operators dual to the Konishi multiplet.