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Chillingworth confirms that he is Hester's long lost husband (we could have figured that one out!)
Chillingworth says he is here not to re-unite with Hester but to ferret out this unnamed man who slept with his wife.
I don't think Chillingworth is really after Hester. He simply wants to ruin the man responsible for making him feel like the pathetic jilted husband. Chillingworth needs to validate himself by causing the prolonged and twisted suffering of another man (Dimmesdale).