The bosonic part of M-(atrix) theory, which is also called the BFSS matrix model, is given explicitly by the action
\begin{eqnarray}
S=\frac{1}{g^2}\int_0^{\beta}dt Tr\bigg[\frac{1}{2}(D_t\Phi_i)^2-\frac{1}{4}[\Phi_i,\Phi_j]^2\bigg]~,~D_t=\partial_t-i[A_t,..].\label{BFSST}
\end{eqnarray}
The fields are periodic with period $\beta=1/T$ where $T$ is the Hawking temperature and we will study the system in the t'Hooft limit given by
\begin{eqnarray}
\lambda=g^2N.
\end{eqnarray}By dimensional analysis we find that $\Phi_i$ behaves as inverse length, $\lambda$ behaves as inverse length cubed and $T$ behaves as inverse length. As a consequence the dimensionless coupling constant must be given by $\tilde{\lambda}={\lambda}/{T^3}$. We will choose $\lambda=1$ so $g^2=1/N$ and the model depends then on only one parameter which is the temperature $T$.
The energy of this matrix model is defined by
\begin{eqnarray}
\frac{E}{N^2}=\frac{3T}{N^2}\langle {\rm commu}\rangle~,~{\rm commu}=-\frac{1}{4g^2}\int_{0}^{\beta}dt{Tr}[{\Phi}_i^{},{\Phi}_j^{}]^2.
\end{eqnarray}
The corresponding radius or extent of space $R^2$ is another very important observable in this model which is given explicitly by
\begin{eqnarray}
R^2=\frac{a}{\Lambda N^2}\langle {\rm radius}\rangle~,~{\rm radius}=\frac{N}{a}\sum_{n=1}^{\Lambda}{Tr}{\Phi}_i^{2}(n).
\end{eqnarray}
The Polyakov line (which acts as our macroscopic order parameter) is defined in terms of the holonomy matrix $U$ (or Wilson loop) by the relation
\begin{eqnarray}
P=\frac{1}{N}Tr U~,~U={\cal P}\exp(-i\int_0^{\beta}dt A(t)).
\end{eqnarray}
After gauge-fixing on the lattice (we choose the static gauge $A(t)=-(\theta_1,\theta_2,...,\theta_N)/\beta$) we write the Polyakov line $P$ in terms of the holonomy angles $\theta_a$ as
\begin{eqnarray}
P=\frac{1}{N}\sum_a\exp(i\theta_a).
\end{eqnarray}
We actually measure in Monte Carlo simulation the expectation value
\begin{eqnarray}
\langle |P|\rangle=\int d\theta \rho(\theta)\exp(i\theta).
\end{eqnarray}
The eigenvalue distribution $\rho(\theta)$ of the holonomy angles is our microscopic order parameter used to characterize precisely the various phases of this model. This eigenvalue distribution is given formally by
\begin{eqnarray}
\rho(\theta)=\frac{1}{N}\sum_{a=1}^N \langle\delta(\theta-\theta_a)\rangle.
\end{eqnarray}
The phase diagram of this model was determined numerically by means of the Monte Carlo method in \cite{Kawahara:2007fnF} to be consisting of two phase transitions and three stable phases.
- The confinement/deconfinement phase transition: At low temperatures the $U(1)$ symmetry $A(t)\longrightarrow A(t)+C.{\bf 1}$ is unbroken and hence we have a confining phase characterized by a uniform eigenvalue distribution. The $U(1)$ symmetry gets spontaneously broken at some high temperature $T_{c 2}$ and the system enters the deconfining phase which is characterized by a non-uniform eigenvalue distribution. This is a second order phase transition.
- The Eguchi-Kawai reduction: Thus, the energy/radius in the confining uniform phase is constant which is consistent with the so-called Eguchi-Kawai equivalence \cite{Eguchi:1982nmF}. This states that the expectation values of single-trace operators in $d-$dimensional large $N$ gauge theories are independent of the volume if the $U(1)^d$ symmetry is not spontaneoulsy broken. In our case $d=1$ and independence of the volume is precisely independence of the temperature which is the inverse Euclidean time.
The constant value of the energy in the confining uniform phase is identified with the ground state energy. The energy in the deconfining non-uniform phase ($T> T_{c2}$) deviates from this constant value quadratically, i.e. as $(T-T_{c 2})^2$.
We can also check that the Polyakov line $\langle |P|\rangle$ vanishes identically in a uniform eigenvalue distribution. But numerically it is observed that $\langle |P|\rangle$ vanishes only as $1/N$ at low temperatures. - The Gross-Witten-Wadia phase transition: This is a third order phase transition occurring at a temperature $T_{c1}> T_{c2}$ dividing therefore the non-uniform phase into two distinct phases: The gapless phase in the intermediate region $T_{c 2}\le T\le T_{c 1}$ and the gapped phase at high temperatures $T> T_{c1}$. The terminology "gapless" means that the eigenvalue distribution has no gaps on the circle whereas "gapped" means that the distribution is non-zero only in the range $[-\theta_0,\theta_0]$. This transition is well described by the Gross-Witten-Wadia one-plaquette unitary model \cite{Gross:1980heF,Wadia:1980cpF}
\begin{eqnarray}
Z_{GWW}=\int dU \exp(\frac{N}{\kappa}Tr U+{\rm h.c}).
\end{eqnarray}
The deconfined non-uniform gapless phase is described by a gapless eigenvalue distribution (and hence the name: gapless phase) of the form
\begin{eqnarray}
\rho_{\rm gapless}=\frac{1}{2\pi}(1+\frac{2}{\kappa}\cos\theta)~,~-\pi<\theta\le+\pi~,~\kappa\ge 2.
\end{eqnarray}
This solution is valid only for $\kappa\ge 2$ where $\kappa$ is a function of the temperature.
At $\kappa=2$ (corresponding to $T=T_{c1}$) a third order phase transition occurs to a gapped eigenvalue distribution given explicitly by
\begin{eqnarray}
\rho_{\rm gapped}=\frac{1}{\pi\sin^2\frac{\theta_0}{2}}\cos\frac{\theta}{2}\sqrt{\sin^2\frac{\theta_0}{2}-\sin^2\frac{\theta}{2}}~,~-\theta_0\le\theta\le+\theta_0~,~\kappa< 2.
\end{eqnarray}
The eigenvalue distribution is non-zero only in the range $[-\theta_0,\theta_0]$ (arbitrarily chosen to be centered around $0$ for simplicity) where the angle $\theta_0$ is given explicitly by
\begin{eqnarray}
\sin^2\frac{\theta_0}{2}=\frac{\kappa}{2}.
\end{eqnarray}
This is a gapped distribution since only the interval $[-\theta_0,\theta_0]$ is filled. At high temperatures corresponding to $\kappa\longrightarrow 0$ the above distribution approaches a delta function \cite{Aharony:2003sxF}. - The $1/d$ expansion: By using a $1/d$ expansion around the $d=\infty$ saddle point of the model (\ref{BFSST}) which is characterized by a non-zero value of the condensate $\langle Tr\Phi_i\Phi_i\rangle$ we can show explicitly that the phase structure of the model consists of two phases: $1)$ a confinement/deconfinement second order phase transition marking the onset of non-uniformity in the eigenvalue distribution closely followed by $2)$ a GWW third order phase transition marking the onset of a gap in the eigenvalue distribution \cite{Mandal:2009vzF,Mandal:2011hbF}.
- Hagedorn transition:
It has been argued that the deconfinement phase transition in gauge theory such as the above discussed phase transition is precisely the Hagedorn phase in string theory \cite{Aharony:2003sxF}. It has also been argued there that the Hagedorn transition could be a single first order transition and not a deconfinement second order transition followed by a gapped third order transition, i.e. the gapless phase may not be there (recall that its range is very narrow). - The black-string/black-hole transition: The confinement/deconfinement phase transition observed in this model, which is the analogue of the confinement/deconfinement phase transition of ${\cal N}=4$ supersymmetric Yang-Mills theory on ${\bf S}^3$, is the weak coupling limit of the black-string/black-hole phase transition observed in the dual garvity theory of two-dimensional Yang-Mills theory.
Thus, the phase structure of gauge theory in one dimension can also be obtained from considerations of holography and the dual gravitational theory (beside and supplementing the Monte Carlo method and the analytical $1/d$ and $1/N$ expansions). In particular, the thermodynamics of a given phase of Yang-Mills gauge theory can be deduced from the Bekenstein-Hawking thermodynamics of the corresponding charged black (string or hole) solution \cite{Aharony:2004igF}.
The dimensionless parameters of the two-dimensional super Yang-Mills gauge theory on the torus ${\bf T}^2$ are $\tilde{T}\tilde{L}$ and $\tilde{\lambda}\tilde{L}^2$ where $\tilde{\beta}=1/\tilde{T}$ and $\tilde{L}$ are the circumferences of the two cycles of ${\bf T}^2$.
At strong 't Hooft coupling $\tilde{\lambda}\longrightarrow\infty$ and small temperature $\tilde{T}$ it was shown in \cite{Aharony:2004igF} that the above $2-$dimensional Yang-Mills theory exhibits a first order phase transition at the value
\begin{eqnarray}
\tilde{T}\tilde{L}=\frac{2.29}{\sqrt{\tilde{\lambda}\tilde{L}^2}}.
\end{eqnarray}
This corresponds in the dual gravity theory side to a transition between the black hole phase (gapped phase) and the black string phase (the uniform and gapless phases) \cite{Susskind:1997drF}. This black-hole/black-string first order phase transition is the Gregory-Laflamme instability in this case \cite{Gregory:1993vyF}.
At large number of dimensions $d\longrightarrow\infty$ the BFSS matrix model is equivalent to a gauged matrix harmonic oscillator problem given by the action \cite{Mandal:2009vzF,Mandal:2011hbF}
\begin{eqnarray}
S[\Phi]=\frac{1}{g^2}\int_0^{\beta} dt Tr\bigg[\frac{1}{2}(D_t\Phi_i)^2+\frac{1}{2}m^2(\Phi_i)^2\bigg]~,~m=d^{1/3}.\label{gauss}
\end{eqnarray}
An analytic study of the Gaussian model (\ref{gauss}) is given in the very interesting paper \cite{Furuuchi:2003syF} where its relevance to the plane wave matrix model and string theory is discussed at length.
Furthermore, it has been argued in \cite{Filev:2015hiaF} that the dynamics of the bosonic BFSS model is fully dominated by the large $d$ behavior encoded in the above quadratic action.
This has been in fact checked in Monte Carlo simulations where a Hagedorn transition is observed consisting of a second order confinement/deconfinement phase transition closely followed by a GWW third order transition which marks the emergence of a gap in the eigenvalue distribution.
Two further remarks are in order here. First, the energy in the present model is given effectively by the extent of space. Indeed, we compute the expression
\begin{eqnarray}
\frac{E}{N^2}=\frac{a^2Tm^2}{N^{2}}\langle{\rm radius}\rangle=m^2R^2.
\end{eqnarray}
Second, in this Gaussian approximation it is observed that the eigenvalues of the adjoint scalar fields $\Phi_i$ are distributed according to the Wigner semi-circle law with a radius $r$ which follows the temperature behavior of the extent of space $R^2$ since $r^2=4R^2/d$. Thus, only the radius of the eigenvalue distribution undergoes a phase transition but not in its shape (which is always given by a Wigner semi-circle law). We also note that at low temperature this radius becomes constant given by $r=\sqrt{2/m}$.
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