BFSS Matrix Theory

String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History	S-Matrix Theory Regge Trajectory
Bosonic String Theory	Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation
Supersymmetric Revolution	Supersymmetry RNS Formalism GS Formalism BPS
Superstring Revolutions
First Superstring Revolution	GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory
Second Superstring Revolution	T-Duality D-Brane S-Duality Horava-Witten String Theory M-Theory Holographic Principle N=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism
After the Revolutions
Phenomenology	String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology

BFSS Matrix Theory, also known as M(atrix) Theory is a fully non-peturbative formulation of M-Theory. It was proposed by Banks, Fischler, Shenker, and Susskind in 1996 .

M(atrix) Theory relies on the AdS/CFT Correspondence, specifically, that

M-Theory in Anti-de-Sitter Space is equivalent to the $N\to \infty$ limit of Supersymmetryic Quantum Yang-Mills Theory describing D0 branes (point particles).

Here, $N$ is the dimension of the gauge group $U(N)$ of the supersymmetric quantum Yang-Mills Theory.

Lagrangian Density

The Lagrangian Density of M(atrix) Theory can immediately be deduced to be the same as in Supersymmetryic Quantum Yang-Mills Theory (N=4 Super-Yang-Mills Theory):

${\mathsf {\mathcal {L}}}={\frac {1}{2{{g}_{s}}\ell _{P}^{2}}}\left(\left(\operatorname {tr} {\frac {{\text{d}}{{X}^{\mu }}}{{\text{d}}\tau }}\right){\frac {{\text{d}}{{X}^{\mu }}}{{\text{d}}\tau }}+2\psi _{\mu }^{}{\frac {{\text{d}}\psi _{\mu }^{}}{{\text{d}}\tau }}-{\frac {1}{2}}{{\left(\operatorname {tr} \left[{{X}^{\mu }},{{X}^{\nu }}\right]\right)}^{2}}-2\psi _{\mu }^{}{{\gamma }_{\mu }}\left[\psi _{}^{\mu },{{X}^{\mu }}\right]\right)$

It can be shown that this gives rise to many expected properties of M-Theory, such as brane tension, and $\mathcal N=8$ Supersymmetry.