A man at the center of a standoff in New Britain died Tuesday morning after a stun gun was used to end the stalemate, State Police said. As officers brought the man down the stairs of the apartment building where the standoff began Monday night, he had difficulty breathing and suffered an unspecified ‘‘medical issue,’’ State Police said. He was declared dead at New Britain General Hospital at 2 a.m. Tuesday. The man’s name was not released pending family notification. State Police were called to the building after receiving reports of an unruly person. The man refused to surrender, a struggle followed, and an officer was forced to use his Taser, State Police said.