Yes, all of the campaigns feature a cutscene once you complete all of the Hegemony Objectives. The Archidamiam War has a fairly long objective chain at the end where you need to complete each objective before it will unlock the next one. I think the first objective in that chain is 'Ravage The Megarid' which unlocks 'A Plague Upon Thee' which then leads to the final 'Dawn of Demosthenes' objectives.

If you open the in-game manual and click 'Objectives' and then 'Hegemony Objectives' you'll get a list of all of the objectives required to finish the campaign. You can also zoom out so you can see the entire map and then click on the sword and wreath icons to see all of the objectives in the campaign.