Shows originally developed at the American Repertory Theater and Huntington Theatre Company will make the leap to New York’s Second Stage Theatre next season, a reminder that the road between Boston and New York is a two-way street. “Witness Uganda” (now retitled “Invisible Thread”), the musical by Matt Gould and Griffin Matthews that premiered at ART in Cambridge last year, will open Second Stage’s 37th season in the fall, the theater announced Thursday. ART artistic director Diane Paulus will direct, as she did for the first run of the show, an autobiographical tuner about a young American’s trials and triumphs teaching schoolchildren in that East African country. And spring 2016 will see the New York premiere of “Smart People,” a think piece about US race relations circa 2008 that premiered at the Huntington last June. Lydia R. Diamond’s play looks at race through the lens of four Cambridge residents, each of whom has ties to Harvard. It will be directed by Tony winner Kenny Leon. Second Stage is a theater company on the move; last month it bought Broadway’s Helen Hayes Theatre, with plans to start producing there in 2016.Jeremy D. Goodwin can be reached at email@example.com.