Relating Berkovits and A_∞ superstring field theories; small Hilbert space perspective

In a previous paper it was shown that the recently constructed action for open superstring field theory based on A_∞ algebras can be re-written in Wess-Zumino-Witten-like form, thus establishing its relation to Berkovits' open superstring field theory. In this paper we explain the relation between these two theories from a different perspective which emphasizes the small Hilbert space, and in particular the relation between the A_∞ structures on both sides.