BFSS Mtheory Equivalence

In the source’s flat eleven-dimensional light-cone convention, let $x^0$ be time, $x^{11}$ the chosen longitudinal coordinate, take metric signature $(-,+,\dots ,+)$, and define $x^{\pm }=(x^0\pm x^{11})\mathrm{/}\sqrt{2}$ with light-front time $t=x^{+}$. Discrete light-cone quantization (DLCQ) means imposing a periodic identification of $x^{-}$ and restricting to a sector in which its conjugate longitudinal momentum is quantized.

For each $N\in {\mathbb{Z}}_{>0}$, let $M_N(\mathbb{C})$ denote the complex $N\times N$ matrices and $U(N)=\{U\in M_N(\mathbb{C}):U^{†}U=I_N\}$. Hermitian means $M^{†}=M$. An adjoint matrix transforms by $M\mapsto UM{U}^{-1}$. The transverse rotation group is $SO(9)=\{O\in M_9(\mathbb{R}):O^{\mathsf{T}}O=I_9,\det O=1\}$; choose real symmetric $16\times 16$ Clifford matrices satisfying ${\gamma }_i{\gamma }_j+{\gamma }_j{\gamma }_i=2{\delta }_{ij}{I}_{16}$. For $u\in {\mathbb{R}}^9$, put $\gamma (u)={\sum }_i{u}_i{\gamma }_i$ and define $\mathrm{Spin}(9)\subset G{L}_{16}(\mathbb{R})$ as the group of products $\gamma (u_1)\cdots \gamma (u_{2m})$, where $m\ge 0$ and every $u_j$ has Euclidean norm one (the empty product is $I_{16}$). Its spin representation on ${\mathbb{R}}^{16}$ is matrix multiplication; its vector action is the rotation $O$ determined by $S\gamma (v)S^{-1}=\gamma (Ov)$ for $S\in \mathrm{Spin}(9)$ and $v\in {\mathbb{R}}^9$. Grassmann-odd components anticommute.

For a fixed positive integer $N$ and $R_{-}>0$, let $x^{-}\sim x^{-}+2\pi R_{-}$ be the lightlike-circle identification and let the compact longitudinal momentum be $P^{+}=N\mathrm{/}{R}_{-}$. The string-unit type-IIA presentation uses $\mathrm{\hslash }=c={\mathrm{\ell }}_s=1$; let the constant asymptotic dilaton ${\mathrm{\Phi }}_{\mathrm{\infty }}\in \mathbb{R}$ define the dimensionless coupling $g_s=e^{{\mathrm{\Phi }}_{\mathrm{\infty }}}>0$. The source's relation $R_{11}=g_s{\mathrm{\ell }}_s$ is for the spatial M-theory circle, whereas $R_{-}$ is the radius of the lightlike circle. In the Seiberg–Sen interpretation of the finite-$N$ DLCQ sector, a separate spacelike regulator radius $R_s$ tends to zero. Here ${\mathrm{\ell }}_{11}$ is the target eleven-dimensional Planck length and ${\tilde{\mathrm{\ell }}}_{11}$ is the rescaled Planck unit used for the spacelike regulator. The source's scaling relation is $R_s\mathrm{/}{\tilde{\mathrm{\ell }}}_{11}^2=R_{-}\mathrm{/}{\mathrm{\ell }}_{11}^2$; the regulator's IIA coupling and string length satisfy ${\tilde{g}}_s=(R_s\mathrm{/}{\tilde{\mathrm{\ell }}}_{11}{)}^{3\mathrm{/}2}\to 0$ and ${\tilde{\mathrm{\ell }}}_s^2={\tilde{\mathrm{\ell }}}_{11}^3\mathrm{/}{R}_s$. For the D0-brane quantum mechanics written in the string-unit variables below, $t$ and $X^i$ denote their dimensionless numerical values and the coupling in its action is $g_s$. Its use as a finite-$N$ DLCQ model is the stated limiting interpretation, not an equality between that action coupling and $R_{-}$. With light-front time $t=x^{+}$, the model is the quantum mechanics with nine Hermitian adjoint matrices $X^i(t)$, $i=1,\dots ,9$, sixteen real Grassmann-odd adjoint matrices ${\mathrm{\Theta }}_{\alpha }(t)$ transforming in the spin representation of $\mathrm{Spin}(9)$, and a nondynamical Hermitian $U(N)$ gauge field $A_0(t)$. With ${\gamma }_i$ the real $16\times 16$ Euclidean Clifford matrices, $D_t{X}^i={\dot{X}}^i-i[A_0,X^i]$, and $D_t\mathrm{\Theta }=\dot{\mathrm{\Theta }}-i[A_0,\mathrm{\Theta }]$, its string-unit action in the displayed Hermitian-matrix convention is

Here $\mathrm{Tr}$ is the fundamental matrix trace. Variation of $A_0$ imposes the $U(N)$ Gauss law: physical wavefunctions are invariant under simultaneous conjugation $X^i\mapsto U{X}^i{U}^{-1}$, $\mathrm{\Theta }\mapsto U\mathrm{\Theta }{U}^{-1}$ for every $U\in U(N)$; equivalently every infinitesimal gauge generator annihilates a physical state. At finite $N$ and $R$, the proposed target is this fixed-momentum DLCQ sector, not all states of uncompactified M-theory. The uncompactified comparison requires a specified scaling of $N$ and $R$; for scattering amplitudes the source takes $N_i,R_{-}\to \mathrm{\infty }$ with each $P_i^{+}=N_i\mathrm{/}{R}_{-}$ fixed, and separately specifies transverse-circle limits when present. These limits and the admissible backgrounds and observables are part of the conjecture, not consequences of the finite-$N$ action.

Establish that large-N BFSS U(N) supersymmetric matrix quantum mechanics is equivalent to M-theory in the infinite-momentum/light-cone frame for the declared backgrounds and observables, including finite-N meaning, scattering, compactification and brane sectors. BFSS is a nonperturbative proposal with tests, not proof of M-theory existence or equivalence.

Banks, Fischler, Shenker, and Susskind, §1, §3 Eq. (3.1), §4 Eq. (4.1), and §§8–9

Taylor, The M(atrix) model of M-theory, Introduction and §§3.2–3.5, especially §3.5

Herderschee and Maldacena, §§2–4

The source-status review dated 2026-10-30 finds this equivalence unresolved. Banks–Fischler–Shenker–Susskind propose the large-$N$ equivalence and give evidence rather than a proof; their discussion leaves questions about the existence and interpretation of the limit, eleven-dimensional Lorentz invariance, and compactification and brane sectors. Taylor distinguishes the finite-$N$ matrix quantum mechanics and its proposed DLCQ-sector interpretation from the unproved equivalence of its limiting sector with complete M-theory. Herderschee and Maldacena’s selected flat-space three-graviton computation uses nonzero longitudinal momenta and a compactified-index setup with one imaginary transverse momentum; its finite-$N$ agreement tests one protected amplitude and does not establish the full equivalence for other observables or backgrounds. This is a targeted source review through 2026-10-30, not an exhaustive literature claim.