Notes on the Wess-Zumino-Witten-like structure: $L_{\infty }$ triplet and NS-NS superstring field theory

In the NS-NS sector of superstring field theory, there potentially exist three nilpotent generators of gauge transformations and two constraint equations: It makes the gauge algebra of type II theory somewhat complicated. In this paper, we show that every NS-NS actions have their WZW-like forms, and that a triplet of mutually commutative $L_{\infty }$ products completely determines the gauge structure of NS-NS superstring field theory via its WZW-like structure. We give detailed analysis about it and present its characteristic properties by focusing on two NS-NS actions proposed by arXiv:1512.03379 and arXiv:1403.0940.