Fully Off-shell Effective Action and its Supersymmetry in Matrix Theory II

In a previous work, we computed the fully off-shell effective action $\Gamma$ and the corresponding quantum-corrected supersymmetry (SUSY) transformation operator $\delta_\epsilon$ for the so-called source-probe configuration in Matrix theory at one loop at order 4 in the derivative expansion, and showed that they satisfy the SUSY Ward identity $\delta_\epsilon \Gamma=0$. In this article, starting from the most general form of $\Gamma$, we demonstrate that, conversely, given such $\delta_\epsilon$ the SUSY Ward identity determines $\Gamma$ uniquely to the order specified above. Our demonstration does not require the explicit knowledge of the quantum-corrected supersymmetry transformation and hence strongly suggests that the uniqueness property would persist to all orders in perturbation theory.