What capital sins are presented in the story?
I think by capital sins you mean which of the seven deadly sins?
sloth --The party guests are locking themselves away to simply party out the outbreak of plague. Instead of creating an environment that is sealed off and productive, they are reveling in lazy pursuits.
gluttony -- The prince has stored away opulent foods and delights enough to last out the outbreak. He and his guests are being gluttons for feasting beyond what is necessary.
pride -- The prince is so proud of himself that he feels that he can outsmart death, illness, and by that matter God.