Our master geodatabase, on SQL Server, contains data layers that are maintained on a separate file geodatabase, and are replicated on the master database. We would like to create attribute indexes on those layers in our master geodatabase. The question is, will synchronization and replication break or cause problems with those attribute indexes when the data updates are pushed from the file geodatabase?
No they shouldn't cause any issues.Mainly it just speeds up the querying process, without altering the schema or data.
However, recommendation is to always test on a small subset of data before implementing in production environment.
You should be fine having attribute indexes in your master geodatabase and using replication with your file geodatabases. The help docs for Importing a replica schema—ArcGIS Help | ArcGIS Desktop indicate that attribute indexes are ignored during schema comparison:
Some changes can be applied to individual replica geodatabases but have no effect on data synchronization. For example, adding or removing an attribute index is not detected or even synchronized during schema synchronization. A replica may have indexes that the relative replica doesn't have, but since data synchronization is not affected, this difference is ignored.
It's probably a good idea to use the Rebuild Indexes tool after synchronizing your replicas for performance reasons. See Rebuild dataset indexes using the Rebuild Indexes tool—ArcGIS Help | ArcGIS Desktop.