Matrix Theory Three Point Amplitude 2026

Herderschee and Maldacena report a match between the compactified eleven-dimensional momentum-space three-supergraviton amplitude ${\mathcal{A}}_3$ (their Eq. (20)) and the canonical-normalized BFSS $2\to 1$ matrix-model S-matrix element ${\mathcal{A}}_{MM}^c$. Let $R_{-}>0$ be the radius of the longitudinal lightlike circle $x^{-}$ and $R_9>0$ the radius of the additional compact transverse circle $x^9$. Label particles 1 and 2 incoming and particle 3 outgoing by integer quantum numbers $N_i>0$ and $n_i\in \mathbb{Z}$, with $x^{-}\sim x^{-}+2\pi R_{-},x^9\sim x^9+2\pi R_9,-p_{-}^i=\frac{N_i}{R_{-}},p_9^i=-i{p}_8^i=\frac{n_i}{R_9},$ and conservation is imposed by $N_3=N_1+N_2$ and $n_3=n_1+n_2$ The all-incoming convention negates particle 3’s momentum, so the conserved sum is $p^1+p^2-p^3=0$ in these physical in/out labels. The $x^8$ direction is noncompact, but the selected on-shell three-point kinematics analytically continue its momentum to $p_8^i=i{n}_i\mathrm{/}{R}_9$; this is not a real-momentum configuration. The source’s canonical external-state normalization is in Eq. (12), and the compactified eleven-dimensional three-point amplitude in that convention is Eq. (20). Equation (21) is the BFSS action, while Eq. (24) is the compactified $1+1$-dimensional matrix-theory action. Its relative-sector boundary state is defined in Eq. (48); the final matrix-model amplitude is Eq. (61), which the source reports agrees precisely with Eq. (20). The boundary state enters the index computation used to obtain that matrix element.

The reported match holds at finite $N_i$ and $n_i$ in the source’s asymptotic extraction: $x_r^8\to \mathrm{\infty }$ and $\mathrm{\Delta }\tau ={\tau }_r-{\tau }_l\to \mathrm{\infty }$ with $x_r^8\mathrm{/}\mathrm{\Delta }\tau$ fixed, where $x_r^8$ is the relative transverse boundary separation and $[{\tau }_l,{\tau }_r]$ is the Euclidean-time interval. These quantities have units of length in $\mathrm{\hslash }=c=1$ units. The boundary-state computation uses gapped external sectors with nonzero $n_i$ and $\gcd (N_i,\mathrm{\mid }{n}_i\mathrm{\mid })=1$, and a nonzero relative flux $(n_1{N}_2-n_2{N}_1)\mathrm{/}{N}_3$; it supplies no independent derivation here for degenerate sectors. The source then takes $R_9\to \mathrm{\infty }$ and $n_i\to \mathrm{\infty }$ with $n_i\mathrm{/}{R}_9$ fixed to recover the $0+1$-dimensional matrix-model amplitude, and takes $N_i\to \mathrm{\infty }$ with $N_i\mathrm{/}{R}_{-}$ fixed to reach the amplitude with noncompact $x^{-}$. This is one protected three-point test, not proof of full BFSS or M-theory equivalence.

Herderschee and Maldacena, §§2–4, equations (8), (12), (20), (21), (24), (48), and (61)

Banks, Fischler, Shenker, and Susskind, §§1 and 5

The source-status review dated 2026-10-30 confirms the reported scope above: all three longitudinal momenta are nonzero, and the selected compactified index calculation uses an analytic continuation with one imaginary transverse momentum. Agreement at finite $N_i,n_i$ after this asymptotic extraction and in the source’s stated noncompact limits tests this one protected amplitude only; it does not establish amplitudes for arbitrary backgrounds or kinematics, nonprotected sectors, brane dynamics, or the full BFSS/M-theory equivalence. This is a report of the source’s calculation, not an independently reconstructed amplitude proof, an experiment, or a general theorem.

This is a reported source calculation, not an independently reconstructed amplitude proof, an experiment, or a general theorem.