Localization of the BFSS matrix model and three-point amplitude in M-theory

We apply the localization method to the BFSS matrix model with a particular class of boundary conditions, that is related to a scattering problem of 11-dimensional M-theory. For the boundary condition that corresponds to the three-point amplitude of gravitons, we exactly compute the partition function of the model based on the localization method. We find that the result correctly reproduces the expected momentum dependence of the three point amplitude.

💡 Research Summary

The paper presents a direct computation of the three‑point graviton amplitude of eleven‑dimensional M‑theory from the BFSS matrix model using supersymmetric localization. The authors first reformulate the BFSS model on a finite line segment, introducing appropriate boundary terms that preserve a single off‑shell supersymmetry corresponding to the 1/4‑BPS sector relevant for the three‑point process. By selecting a specific Killing spinor ϵ (constructed in the appendix) they obtain four dynamical supersymmetries on the boundary, which can be combined with a BRST symmetry to define a nilpotent operator Q = δ – δ_B.

To make the action Q‑exact, they add a cubic Myers‑type boundary term S_b that cancels the supersymmetry‑breaking surface contributions arising from the bulk action. The combined bulk‑plus‑boundary action is invariant under Q, allowing the use of localization. The Q‑fixed point equations reduce to the Nahm equation D X^a + (i/2) ε^{abc}