There aren’t many conductors who specialize in ballet repertoire, so when New York City Ballet’s music director announced that she was leaving the company two years ago, officials scrambled to find a replacement who could do more than just give the beat. They found Fayçal Karoui, a flamboyantly energetic Frenchman without much ballet experience, but with an enthusiasm for the work that seems to dovetail nicely with City Ballet’s “unusually broad musical repertory and Balanchine-derived belief in the primacy of the music.”