Phases of gauge theory in lower dimensions and the black-hole/black-string transition

The model

Gauge theory in one dimension with $d$ adjoint scalar fields is intimately related to string theory (for example D branes on tori) and black hole physics (for example the Gregory-Laflamme instability). The action can be obtained from the dimensional reduction of $(d+1)-$dimensional Yang-Mills theory to $1$ dimension. The action is of the form
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].\label{bfssB}
This matrix quantum mechanics is in fact the bosonic part of the  BFSS M-(atrix) theory which corresponds to DLCQ of M-theory and describes D0-branes (which is perhaps the most important connection to string theory and black holes).

The only free parameter in this model is the Hawking temperature $T=1/\beta$. On the lattice with the time direction given by a circle the inverse temperature $\beta$ is precisely equal the circumference of the circle, viz $\beta=a.\Lambda$ where $a$ is the lattice spacing and $\Lambda$ is the number of links.

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
P=\frac{1}{N}Tr U~,~U={\cal P}\exp(-i\int_0^{\beta}dt A(t)).
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
We actually measure in Monte Carlo simulation the expectation value
\langle |P|\rangle=\int d\theta \rho(\theta)\exp(i\theta).
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
\rho(\theta)=\frac{1}{N}\sum_{a=1}^N \langle\delta(\theta-\theta_a)\rangle.
The energy $E/N^2$ of the bosonic truncation of the BFSS matrix model and its corresponding extent of space $R^2$ are the other very important observables in this model. They are given explicitly by
\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.

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).

The phase structure 

The phase diagram of this model was determined numerically by means of the Monte Carlo method in \cite{Kawahara:2007fn} to be consisting of two phase transitions and three stable phases. In more detail, we have

The confinement/deconfinement phase transition: This is a second order phase transition associated with the spontaneous breakdown of the $U(1)$ symmetry
A(t)\longrightarrow A(t)+C.{\bf 1}.
At low temperatures this symmetry is unbroken and as a consequence we have a confining  phase characterized by a uniform eigenvalue distribution, i.e.
\rho(\theta)=\frac{1}{2\pi}~,~T\longrightarrow 0.
As the temperature increases the above $U(1)$ symmetry gets spontaneously broken at some temperature $T_{c 2}$ and we enter the deconfining phase which is characterized by a non-uniform eigenvalue distribution.

We can easily 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 \cite{Kawahara:2007fn}.

The energy $E/N^2$ and the extent of space $R^2$ in the confining uniform phase are both constant which is consistent with the so-called Eguchi-Kawai equivalence \cite{Eguchi:1982nm} (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 ($T\gt T_{c2}$) deviates from this constant value quadratically, i.e. as  $(T-T_{c 2})^2$.  This is confirmed from the behavior of the extent of space $R^2$ which is constant in the  confining uniform phase then deviates from this constant value quadratically as well \cite{Kawahara:2007fn}.

The Gross-Witten-Wadia phase transition:
This is a third order phase transition occurring at a temperature $T_{c1}\gt T_{c2}$ 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 $T\gt T_{c1}$.

It is observed  in numerical simulations \cite{Kawahara:2007fn} that this phase transition is well described by the Gross-Witten-Wadia one-plaquette model given explicitly by \cite{Gross:1980he,Wadia:1980cp}
Z_{GWW}=\int dU \exp(\frac{N}{\kappa}Tr U+{\rm h.c}).
The deconfined non-uniform gapless phase is described by a gapless eigenvalue distribution (and hence the name: gapless phase) of the form
\rho_{\rm gapless}=\frac{1}{2\pi}(1+\frac{2}{\kappa}\cos\theta)~,~-\pi\lt\theta\le+\pi~,~\kappa\ge 2.
The fact that the angle $\theta$ takes values in the full range $]-\pi,+\pi]$ is precisely what is meant by the word "gapless", i.e. there are no gaps on the circle. 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
\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\lt 2.
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
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:2003sx}.

This third order phase transition is associated therefore with the appearance of a gap in the eigenvalue distribution. We notice that the deconfining non-uniform phase is dominated by the gapped phase since the region of the gapless phase is extremely narrow.

Thus, the Polyakov line suffers another phase transition in the non-uniform phase where it rises from $0$ to the value $1/2$ at $\kappa=2$ ($T=T_{c1}$) in the gapless phase then rises further from $1/2$ to the value $1$ in the gapped phase. We can calculate explicitly the behavior
P=1-\frac{\kappa}{4}~,~\kappa\lt 2.
P=\frac{1}{\kappa}~,~\kappa\ge 2.

Hagedorn transition

It has been argued that the deconfinement phase transition in gauge theory such as the above discussed phase trnasition is precisely the Hagedorn phase in string theory \cite{Aharony:2003sx}. 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).

$1/d$ expansion

The above phase structure was derived in \cite{Mandal:2009vz} in the limit $d\longrightarrow\infty$ (a $1/d$ expansion is considered in lattice models originally in \cite{Drouffe:1979dh,Drouffe:1983fv}). See the related treatment in \cite{Kabat:2000zv,Kabat:2001ve}.

By using a $1/d$ expansion around the $d=\infty$ (here $d=9$ which is reasonably large) saddle point of the model (\ref{bfssB}) 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 (\ref{bfssB}) 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.

The large $d$ limit at finite $N$ is defined,  in analogy with the large $N$ limit, by $d\longrightarrow\infty$ and $g^2\longrightarrow\infty$ keeping $g^2d={\rm  fixed}$. Thus, the large $N$ and large $d$ limit is defined by $\tilde{\lambda}=g^2Nd= \lambda d$ where $\lambda=g^2N$ is 't Hooft limit which can be chosen to be $1$, i.e. $g^2=1/N$. It is found that fluctuations around the saddle point in this limit of large $d$ and large $N$ are suppressed by powers of $1/d$.

In this large $d$ and large $N$ limit, and after the introduction of an $SO(d)-$invariant field $B_{ij}$ to get ride of the Yang-Mills term, the dynamically massive adjoint scalars $\Phi^i$ can be integrated out yielding an effective action depending only on the gauge field $A$. As it turns out this effective action is gauge-invariant since it depends only on the eigenvalues $e^{i\theta_a}$ of the gauge-covariant holonomy matrix or Wilson loop $U$. Explicitly, the effective action is a Landau-Ginzburg energy functional of the form (see \cite{Aharony:2003sx} for the original derivation)
\frac{S}{N^2 d}=\frac{3}{8}\beta\tilde{\lambda}^{1/3}+a_1|u_1|^2+b_1|u_1|^4+\sum_{n=2}a_n|u_n|^2+...
The $u_n$ are the moments of the Wilson loop $U$, viz
u_n=\frac{1}{N}\sum_a\exp(in\theta_a)=\int d\theta \rho(\theta)\exp(in\theta). \label{moments}
The constants $a_n$ and $b_1$ are given by
Obviously, for $\bar{x}\lt 1/d$ all the $a_n$ are positive and thus $u_n=0$ for all $n$ is a global minimum of the potential.  The vanishing of all the $u_n$ corresponds, from (\ref{moments}), to  a uniform eigenvalue distribution and thus this phase is the confinement phase of gauge theory where $TrU=0$.

As the temperature increases $\bar{x}$ crosses the value $1/d$ and as a consequence the quadratic coefficient $a_1$ becomes negative while the rest of the $a_n$ and the $b_n$ remain positive.  The $|u_n|$ for $n\gt 1$ remain zero while $|u_1|$ takes the value $|u_1|=\sqrt{-a_1/2b_1}$ for $T\geq T_{c2}$ where the critical temperature $T_{c2}$  is determined from the condition $a_1=0$ to be given by \cite{Mandal:2009vz}
T_{c2}=\frac{\lambda^{1/3}d^{1/3}}{\ln d}.
This is a second order phase transition (the second derivative of the free energy is found to be discontinuous) which marks the onset of non-uniformity in the eigenvalue distribution.

Above $T_{c2}$ (where all the $|u_n|$ for $n\gt 1$ vanishes) the eigenvalue distribution can be given by
For $|u_1|=0$ this is a uniform distribution whereas for small $|u_1|$ in the region $|u_1|\leq 1/2$ we have a GWW gapless eigenvalue distribution, i.e. a distribution which does not vanish in the whole interval $]-\pi,+\pi]$. As $|u_1|$ increases from $0$ to $1/2$ the above eigenvalue dsitribution vanishes at $\theta=\pi$. Here we enter a new phase characterized by a GWW gapped eigenvalue distribution. The critical temperature $T_{c1}$ can be determined from the condition where the saddle point value   $\langle |u_1|\rangle=\sqrt{-a_1/2b_1}$ reaches the value $1/2$. We get \cite{Mandal:2009vz}
T_{c1}=T_{c2}+\frac{\tilde{\lambda}^{1/3}}{6d\ln d}.
This GWW transition is associated with the onset of a gap in the eigenvalue distribution and it is of a third order  (the third derivative of the free energy is discontinuous) as shown originally in \cite{Aharony:2003sx, AlvarezGaume:2005fv}.

The comparison between the Monte Carlo data of \cite{Kawahara:2007fn} which is performed for $d=9$ adjoint scalars (the dimension of the reduced space is $d=9$) and the $1/d$ analysis around the $d=\infty$ saddle point of \cite{Mandal:2009vz} is very favourable. We get

The $1/d$ expansion can also be applied to gauge theory in $0$ dimension (the infinite temperature limit of the model (\ref{bfssB}) which is essentially the celebarted IKKT model \cite{Ishibashi:1996xs}) where it can be shown that the $1/N$ expansion can be performed exactly \cite{Mandal:2009vz}.  This  expansion in $0$ dimension was actually done originally in \cite{Hotta:1998en}.

Connection to two-dimensional Yang-Mills theory and D-branes description

The connection between the phase structure of low dimensional gauge theories and D-branes (especially  the $D0-$branes and $D1-$branes in the current case) is discussed in great detail  in \cite{Aharony:2004ig,Aharony:2005ew}. See also {Azeyanagi:2009zf,Kawahara:2007fn,Mandal:2009vz}.

In summary, the confinement/deconfinement phase transition observed in the model (\ref{bfssB}), 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 low dimensions can also be obtained from considerations of holography and the dual gravitational theory (beside and supplementing the numerical Monte Carlo and the analytical $1/d$ and $1/N$ expansions considered in previous sections). 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:2004ig}.

Conversely,  the black-string/black-hole phase transition (which is an example of the Gregory-Laflamme instability) can be successfully studied beyond the supergravity approximation using gauge theory methods thanks to the gauge/gravity duality and the AdS/CFT correspondence \cite{Itzhaki:1998dd}.

We start from two-dimensional super Yang-Mills gauge theory on the torus ${\bf T}^2$, i.e. (with $\mu,\nu=0,1$ and $I,J=1,...,8$)
S=\frac{1}{g^2_{\rm YM}}\int d^{2} x\big[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}(D_{\mu}X_I)(D^{\mu}X_I)+\frac{1}{4}[X_I,X_J]^2+\frac{i}{2}\bar{\psi}\gamma^{\mu}D_{\mu}\psi+\frac{1}{2}\bar{\psi}\gamma^{I}[X_{I},\psi]\big].
This is equivalent to type IIB superstring theory around the black 1-brane background spacetime. Thus, we are dealing with a system of $N$ coincident D1-branes forming a black 1-brane solution. This black 1-brane solution can be mapped via S-duality to a black string solution winding around the circle in ${\bf R}^8\times{\bf S}^1$ \cite{Itzhaki:1998dd}.

The dimensionless parameters of this model 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:2004ig} that the  above $2-$dimensional Yang-Mills theory exhibits a first order phase transition at the value
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:1997dr}. This black-hole/black-string first order phase transition is the Gregory-Laflamme instability in this case \cite{Gregory:1993vy}.

Indeed, for very large compactification circumference $\tilde{L}$ the two-dimensional super Yang-Mills gauge theory describes $N$ coincident $D1-$branes in type IIB string theory which are winding on the circle. When $\tilde{L}\longrightarrow 0$ the appropriate description becomes given by $T-$duality \cite{Taylor:1996ik} in terms of $N$ coincident $D0-$branes in type IIA string theory which are winding on a circle  of circumference $\alpha^{\prime}/\tilde{L}$.

The positions of these $D-$particles on the $T-$dual circle are given by the eigenvalues of the Wilson loop winding on the circle, i.e. of the holonomy matrix
W={\cal P}\exp(i\oint dx A_x).\label{Wl}
By an appropriate gauge transformation the Wilson line can be diagonalized as
W={\rm diag}(\exp(i\theta_1),...,\exp(i\theta_N)).
The phase $\theta_a$ is precisely the position of the $a$th $D0-$brane on the $T-$dual circle. If all the angles $\theta_a$ accumulate at the same point then we obtain a black hole at that location whereas if they are distributed uniformly on the circle we obtain a uniform black string. We can also obtain a non-uniform black string phase or a phase with several black holes depending on the distribution of the eigenvalues $\theta_a$.

In the high temperature limit $\tilde{T}\longrightarrow 0$ and weak 't Hooft coupling $\tilde{\lambda}\longrightarrow 0$ the above two-dimensional super Yang-Mills gauge theory on the torus ${\bf T}^2$ reduces to the one-dimensional bosonic gauge theory (\ref{bfssB}).

The Wilson loop (\ref{Wl}) winding around the spatial circle in the $2-$dimensional theory becomes in the $1-$dimensional theory the Polyakov line $P=TrU/N$ since the time direction of this $1-$dimensional model is the spatial direction of the $2-$dimensional model.  More precisely, we have
The high temperature $2-$dimensional Yang-Mills theory on a circle was studied in \cite{Aharony:2004ig,Harmark:2004ws} where a phase transition around $\lambda_{\rm eff}=1.4$ was observed. This result was made more precise by studying the $1-$dimensional matrix quantum mechanics in \cite{Kawahara:2007fn} where two phase transitions were identified of second and third order respectively at the values (see next section for detailed discussion)

\lambda_{\rm eff}=1.35(1)\Rightarrow \tilde{T}\tilde{L}=\frac{1.35(1)}{\tilde{\lambda}\tilde{L}^2}.
\lambda_{\rm eff}=1.487(2)\Rightarrow \tilde{T}\tilde{L}=\frac{1.487(2)}{\tilde{\lambda}\tilde{L}^2}.
The second order transition separates between the gapped phase and the non-uniform phase whereas the third order separates between the non-uniform phase and the uniform phase. These phases, in the $2-$dimensional phase diagram with axes  given by the dimensionless parameters $\tilde{T}\tilde{L}$ and $\tilde{\lambda}\tilde{L}^2$, occur at high temperatures in the region where the $2-$dimensional Yang-Mills theory reduces to the bosonic part of the $1-$dimensional BFSS quantum mechanics. It is  conjectured in \cite{Kawahara:2007fn} that by continuing the above two lines to low temperatures we will reach a triple point where the two lines intersect and as a consequence the non-uniform phase ceases to exist below this  tri-critical point.


It has been argued in \cite{Filev:2015hia} that the dynamics of the bosonic BFSS model (\ref{bfssB}) is fully dominated by the large $d$ behavior encoded in the quadratic action (with $m=d^{1/3}$)
S_{}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2+\frac{1}{2}m^2\Phi_i^2\bigg].\label{gauss}
This has been checked in Monte Carlo simulations where a Hagedorn/deconfinement 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.

In this 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 $r$ following 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 not in its shape (which is always a Wigner semi-circle law). At low temperature this radius becomes constant given by $r=\sqrt{2/m}$.

An analytic study of the model (\ref{gauss}) is given in the beautiful paper \cite{Furuuchi:2003sy} where its relevance to the plane wave matrix model and string theory is discussed at length.

The BMN plane wave gauge theory

The action (\ref{bfssB}) is the bosonic part of the BFSS matrix model which is given by the supersymmetric one-dimensional gauge theory \cite{Banks:1996vh}
S_{\rm BFSS}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2-\frac{1}{4}[\Phi_i,\Phi_j]^2+\frac{1}{2}\psi_{\alpha}D_t\psi_{\alpha}-\frac{1}{2}\psi_{\alpha}(\gamma_i)_{\alpha\beta}[\Phi_i,\psi_{\beta}]\bigg].\label{BFSS}
The only mass deformation that can be added to this action so that maximal supersymmetry is preserved is the plane wave deformation given by the following quadratic mass terms and Chern-Simons  term:

\Delta  S_{\rm defor}&=&\frac{1}{g^2}\int_0^{\beta}dt{Tr}\bigg[\frac{\mu^2}{2}\sum_{a=1}^3\Phi_a^2+\frac{\mu^2}{8}\sum_{i=4}^9\Phi_i^2+i\mu\sum_{a,b,c=1}^3\epsilon_{abc}\Phi_a\Phi_b\Phi_c\bigg]\nonumber\\
Indeed, the celebrated BMN matrix model is given simply by the sum of the above two actions, viz  \cite{Berenstein:2002jq}
S_{\rm BMN}=S_{\rm BFSS}+\Delta  S_{\rm defor}.\label{BMN}
The bosonic truncation of this model gives us then the one-dimensional gauge theory (with $\mu_1=\mu_2=2\alpha/3\equiv \mu$)
S_{}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2-\frac{1}{4}[\Phi_i,\Phi_j]^2+\frac{\mu_1^2}{2}\sum_{a=1}^3\Phi_a^2+\frac{\mu_2^2}{8}\sum_{i=4}^9\Phi_i^2+\frac{2i\alpha}{3}\sum_{a,b,c=1}^3\epsilon_{abc}\Phi_a\Phi_b\Phi_c\bigg].
The phase structure of the model $\mu_1=\mu_2=0$ was studied using the Monte Carlo method in \cite{Kawahara:2007nw}. In particular, it was shown that a fuzzy sphere phase exists above some value $\tilde{\alpha}_c$ of the coupling constant $\tilde{\alpha}=\alpha_cN^{1/3}$ where  the Hagedorn/deconfinement transition is found to be absent. This critical value $\tilde{\alpha}_c$ is a function of the scaled temperature $\tilde{T}=T/N^{2/3}$. Below $\tilde{\alpha}_c$ the Yang-Mills phase is observed  to be divided by the  Hagedorn/deconfinement temperature $T_H$ of the original bosonic BFSS model (the model with $\alpha=0$).

The supersymmetric BMN model (\ref{BMN}) seems however to support the Hagedorn/deconfinement transition (a single phase of of first order) in the fuzzy sphere background \cite{Furuuchi:2003sy, Kawahara:2006hs}.  A first order Hagedorn/deconfinement transition is also expected to appear in the supersymmetric BFSS model (\ref{BFSS})  \cite{Barbon:1998cr, Aharony:2005ew}

It is very interesting to consider also the phase structure of the Gaussian approximation of the above model obtained in the limit $d\longrightarrow \infty$ which is given by (with $m=d^{1/3}$)

S_{}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2+\frac{1}{2}m^2\Phi_i^2+\frac{\mu_1^2}{2}\sum_{a=1}^3\Phi_a^2+\frac{\mu_2^2}{8}\sum_{i=4}^9\Phi_i^2+i\alpha\sum_{a,b,c=1}^3\epsilon_{abc}\Phi_a\Phi_b\Phi_c\bigg].



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