Chris is in conflict with his mother over whether or not he should marry Ann, his brother's former girlfriend. Kate, Chris's mother, insists that Larry is still alive and so it would be wrong for Chris to marry his girlfriend.
Chris, however, is assured that Larry will not be coming home. Ann feels the same way.
The conflict between the two would-be lovers is rather slight. Chris is hesitant to "claim" Ann or to truly embrace her, literally and figuratively. He is worried that he is betraying a truth about himself that he discovered in the war about living authentically.
The conflict between Chris and his father is related to the conflict that animates the play at large. Chris does not want to suspect his father of committing a fraud that led to the deaths of twenty-one pilots, but ultimately comes to understand that Joe Keller is guilty of this crime.
The conflict between the two men is directly drawn from this act of betrayal on Joe's part. Joe is extremely upset by the loss of his son's faith and affection.
He talks, too, of needing Chris's forgiveness and his intent to take his own life should he not get it.