Brown faces fitness race after surgery
The All Whites vice-captain had three screws inserted into the bone he damaged in Monday's match against Australia in Melbourne during the operation in Auckland on Thursday.
"Initial indications are positive and that the surgery went well, but it is too early yet to say what the recovery time frame for the injury is," a statement on the New Zealand Football website said.
Team Wellington midfielder Cole Peverley, 21, has been brought into the 23-man squad as a "standby" for Brown.
New Zealand open their World Cup campaign against Slovakia in Rustenburg on June 15 before facing champions Italy and Paraguay in their other Group F matches.