D-instanton Induced Effective Action and its Gauge Invariance

The effect of D-instantons on closed string scattering amplitudes may be encoded into an effective action obtained by integrating out the (transient) open string fields. In order that this respects the gauge invariance of the theory, the sum of the perturbative closed string field theory action and the D-instanton induced effective action must satisfy the quantum Batalin-Vilkovisky master equation. In a previous paper this was proved for the effective action induced by up to two instantons. We generalize the proof for arbitrary number of D-instantons.