Modes of the Sakai-Sugimoto soliton

The instanton in the Sakai-Sugimoto model corresponds to the Skyrmion on the holographic boundary—which is asymptotically flat—and is fundamentally different from the flat Minkowski space Yang–Mills instanton. We use the Atiyah–Patodi–Singer index theorem and a series of transformations to show that there are 6k zeromodes—or moduli—in the limit of infinite ‘t Hooft coupling of the Sakai-Sugimoto model. The implications for the low-energy baryons—the Skyrmions—on the holographic boundary, is a scale separation between 2k ‘heavy’ massive modes and 6k−9 ‘light’ massive modes for k > 1; the 9 global transformations that correspond to translations, rotations and isorotations remain as zeromodes. For k = 1 there are 2 ‘heavy’ modes and 6 zeromodes due to degeneracy between rotations and isorotations.