Two approaches to quantum gravity and M-(atrix) theory at large number of dimensions

We start then with the bosonic part of the M-(atrix) theory (1.2) at large values of the dimension $d$. The action reads explicitly

$\displaystyle S=\frac{1}{g^{2}}\int_{0}^{\beta}dtTr\bigg{[}\frac{1}{2}(D_{t}\Phi_{i})^{2}-\frac{1}{4}[\Phi_{i},\Phi_{j}]^{2}\bigg{]}.$

(2.1)

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 to the circumference of the circle, viz $\beta=a.\Lambda$ where $a$ is the lattice spacing and $\Lambda$ is the number of links.

The energy of the bosonic truncation (2.1) of the BFSS matrix model (1.2) is defined by (where $Z(\beta)$ is the partition function at temperature $T=1/\beta$)

$\displaystyle E=-\frac{1}{Z(\beta)}\frac{Z(\beta^{{}^{\prime}})-Z(\beta)}{\Delta\beta}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \Delta\beta=\beta^{{}^{\prime}}-\beta\longrightarrow 0.$

(2.2)

We compute immediately [23]

$\displaystyle\frac{E}{N^{2}}=\frac{3T}{N^{2}}\langle{\rm commu}\rangle\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\rm commu}=-\frac{1}{4g^{2}}\int_{0}^{\beta}dt{Tr}[{\Phi}_{i},{\Phi}_{j}]^{2}.$

(2.3)

The corresponding radius or extent of space $R^{2}$ is another very important observable in this model which is given explicitly by

$\displaystyle R^{2}=\frac{a}{\Lambda N^{2}}\langle{\rm radius}\rangle\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\rm radius}=\frac{N}{a}\sum_{n=1}^{\Lambda}{Tr}{\Phi}_{i}^{2}(n).$

(2.4)

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

$\displaystyle P=\frac{1}{N}TrU\leavevmode\nobreak\ ,\leavevmode\nobreak\ U={\cal P}\exp(-i\int_{0}^{\beta}dtA(t)).$

(2.5)

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

$\displaystyle P=\frac{1}{N}\sum_{a}\exp(i\theta_{a}).$

(2.6)

We actually measure in Monte Carlo simulation the expectation value

$\displaystyle\langle|P|\rangle=\int d\theta\rho(\theta)\exp(i\theta).$

(2.7)

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

$\displaystyle\rho(\theta)=\frac{1}{N}\sum_{a=1}^{N}\langle\delta(\theta-\theta_{a})\rangle.$

(2.8)

At high temperatures the bosonic part of the BFSS quantum mechanics reduces to the bosonic part of the IKKT model [22]. The leading behavior of the various observables of interest at high temperatures can be obtained in terms of the corresponding expectation values in the IKKT model. We get then

$\displaystyle R^{2}=\sqrt{T}\chi_{1}.$

(2.9)

$\displaystyle\langle|P|\rangle=1-\frac{1}{2d}{T}^{-3/2}\chi_{1}.$

(2.10)

$\displaystyle\frac{E}{N^{2}}=\frac{3}{4}{T}\chi_{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \chi_{2}=(d-1)(1-\frac{1}{N^{2}}).$

(2.11)

The coefficient $\chi_{1}$ for various $d$ and $N$ can be read off from table $1$ of [22] whereas the coefficient $\chi_{2}$ was determined exactly from the Schwinger-Dyson equation.

The behavior given by equations (2.9), (2.10) and (2.11) can be used to calibrate Monte Carlo simulations at high temperatures.

The phase diagram of this model was determined numerically by means of the Monte Carlo method in [23] 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 $T_{c2}$ and the system enters the deconfining phase which is characterized by a non-uniform eigenvalue distribution. This is a second order phase transition.

• •

  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 $T_{c2}\leq T\leq T_{c1}$ and the gapped phase at high temperatures $T>T_{c1}$¹¹1The 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 [25, 26].

• •

  The $1/d$ expansion: By using a $1/d$ expansion around the $d=\infty$ saddle point of the model (2.1) 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 [35, 36]. See also [31, 32, 29, 30, 21].

• •

  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 [27, 28]. 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: Indeed, 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 [33].

  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 [33] that the above $2-$dimensional Yang-Mills theory exhibits a first order phase transition at the value

  $\displaystyle\tilde{T}\tilde{L}=\frac{2.29}{\sqrt{\tilde{\lambda}\tilde{L}^{2}}}.$

  (2.12)

  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) [34]. This black-hole/black-string first order phase transition is the Gregory-Laflamme instability in this case [12].

For quenched fermions and large number of dimensions $d\longrightarrow\infty$ the BFSS matrix model is equivalent to a gauged matrix harmonic oscillator problem given by the action [35, 36]

$\displaystyle S[\Phi]=\frac{1}{g^{2}}\int_{0}^{\beta}dtTr\bigg{[}\frac{1}{2}(\partial_{t}\Phi_{i})^{2}+\frac{1}{2}m^{2}(\Phi_{i})^{2}\bigg{]}\leavevmode\nobreak\ ,\leavevmode\nobreak\ m=d^{1/3}.$

(2.13)

This gauged matrix harmonic oscillator is a thermal field theory in one dimension and thus $t$ must be an imaginary time, viz $\tilde{t}=-it$ is the real time. As before, 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

$\displaystyle\lambda=g^{2}N.$

(2.14)

Again, there seems to be two independent coupling constants $\lambda$ and $T$. However, $g^{2}$ can always be rescaled away. By dimensional analysis we find that $\Phi_{i}$ behaves as inverse length and $\lambda$ as inverse length cubed whereas $T$ behaves as inverse length and as a consequence the dimensionless coupling constant must be given by $\tilde{\lambda}={\lambda}/{T^{3}}$. We will choose $\lambda=1$, i.e. $g^{2}=1/N$.

It has been argued in [37] 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 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.

First, we must regularize the theory as before using a standard lattice in the Euclidean time direction with a lattice spacing $a$. Recall also that $\beta=\Lambda.a$ is the period on the discrete circle. The lattice gauge-invariant lattice action is straightforwardly given by

$\displaystyle S=\frac{1}{g^{2}}\sum_{n=1}^{\Lambda}{Tr}\bigg{[}\frac{1}{a}\Phi_{i}^{2}(n)-\frac{1}{a}U_{n,n+1}\Phi_{i}(n+1)U_{n+1,n}\Phi_{i}(n)+\frac{1}{2}m^{2}a\Phi_{i}(n)^{2}\bigg{]}.$

(2.15)

The adopted gauge-fixing condition is effectively the so-called static gauge given by [23, 37]

$\displaystyle U_{n,n+1}=D_{\Lambda}={\rm diag}(\exp(i\frac{\theta_{1}}{\Lambda}),...,\exp(i\frac{\theta_{N}}{\Lambda}))\leavevmode\nobreak\ ,\leavevmode\nobreak\ \forall n.$

(2.16)

In terms of the gauge field $A$ this condition reads

$\displaystyle A(t)=-\frac{1}{\beta}{\rm diag}(\theta_{1},...,\theta_{N}).$

(2.17)

In summary, we are interested in the total lattice action (includin the Faddeev-Popov gauge-fixing term)

	$\displaystyle S_{\rm total}$	$\displaystyle=$	$\displaystyle N\sum_{n=1}^{\Lambda}{Tr}\bigg{[}\frac{1}{a}{\Phi}_{i}^{2}(n)+\frac{1}{2}am^{2}{\Phi}_{i}^{2}(n)\bigg{]}$		(2.18)
		$\displaystyle-$	$\displaystyle\frac{N}{a}\sum_{n=1}^{\Lambda}{Tr}{\Phi}_{i}(n)D_{\Lambda}{\Phi}_{i}(n+1)D_{\Lambda}^{\dagger}-\frac{1}{2}\sum_{a\neq b}\ln\sin^{2}\frac{\theta_{a}-\theta_{b}}{2}.$

The Polyakov line is the order parameter of the Hagedorn transition in string theory and the deconfinment transition in gauge theory which is associated with the spontaneous breakdown of the $U(1)$ symmetry $A(t)\longrightarrow A(t)+C.{\bf 1}$. The Polyakov line is defined in terms of the holonomy matrix $U$ by the relation

$\displaystyle P=\frac{1}{N}TrU\leavevmode\nobreak\ ,\leavevmode\nobreak\ U={\cal P}\exp(-i\int_{0}^{\beta}dtA(t)).$

(2.19)

We compute [23]

$\displaystyle U$

$\displaystyle=$

$\displaystyle U_{1,2}U_{2,3}...U_{\Lambda-1,\Lambda}U_{\Lambda,1}=D_{\Lambda}^{\Lambda}={\rm diag}(\exp(i\theta_{1}),...,\exp(i\theta_{N}).$

(2.20)

Hence

$\displaystyle P=\frac{1}{N}\sum_{a}e^{i\theta_{a}}.$

(2.21)

Another important observable is the radius (or extent of space or more precisely the extent of the eigenvalue distribution) which is defined by

$\displaystyle R^{2}=\frac{a}{\Lambda N^{2}}\langle{\rm radius}\rangle\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\rm radius}=\frac{N}{a}\sum_{n=1}^{\Lambda}{Tr}{\Phi}_{i}^{2}(n).$

(2.22)

The energy in the present model is given effectively by the extent of space. Indeed, we compute the energy

$\displaystyle\frac{E}{N^{2}}=\frac{a^{2}Tm^{2}}{N^{2}}\langle{\rm radius}\rangle=m^{2}R^{2}.$

(2.23)

We also measure the eigenvalues distribution of the holonomy matrix $U$ and the eigenvalues distribution of the bosonic matrices $\Phi_{i}(n)$.

We employ the Metropolis algorithm where the variation of the holonomy angles $\theta_{a}$ requires a very careful treatment due to the center of mass $\theta_{\rm cm}=\sum_{a=1}^{N}\theta_{a}/N$ which does not appear in the action and thus behaves as a random walker. See for example [51].

Some of our Monte Carlo results are:

• •

  First, 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.

  See figure (1).

• •

  The extent of space $R^{2}$ (or equivalently the energy) in the confining uniform phase is constant which is consistent with the so-called Eguchi-Kawai equivalence [38]. 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 $(T-T_{c2})^{2}$. This is confirmed in the full bosonic model (2.1) in [23].

  This result is shown on figure (1).

• •

  It is observed in numerical simulations that the second phase transition in the non-uniform phase is well described by the Gross-Witten-Wadia one-plaquette model [25, 26]

  $\displaystyle Z_{GWW}=\int dU\exp(\frac{N}{\kappa}TrU+{\rm h.c}).$

  (2.24)

  The deconfined non-uniform gapless phase is described by a gapless eigenvalue distribution (and hence the name: gapless phase) of the form

  $\displaystyle\rho_{\rm gapless}=\frac{1}{2\pi}(1+\frac{2}{\kappa}\cos\theta)\leavevmode\nobreak\ ,\leavevmode\nobreak\ -\pi<\theta\leq+\pi\leavevmode\nobreak\ ,\leavevmode\nobreak\ \kappa\geq 2.$

  (2.25)

  This solution is valid only for $\kappa\geq 2$ 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

  $\displaystyle\rho_{\rm gapped}=\frac{1}{\pi\sin^{2}\frac{\theta_{0}}{2}}\cos\frac{\theta}{2}\sqrt{\sin^{2}\frac{\theta_{0}}{2}-\sin^{2}\frac{\theta}{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ -\theta_{0}\leq\theta\leq+\theta_{0}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \kappa<2.$

  (2.26)

  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

  $\displaystyle\sin^{2}\frac{\theta_{0}}{2}=\frac{\kappa}{2}.$

  (2.27)

  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 [27].

  A sample of these eigenvalue distributions is shown on figures (2) and (3).

• •

  Also, 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 $r$ 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 Gaussian model (2.13) is given in the very interesting paper [50] where its relevance to the plane wave matrix model and string theory is discussed at length.

For simplicity, we consider Yang-Mills gauge theory in one dimension with just $d=3$ adjoint scalar fields $\Phi_{a}$ with an additional Chern-Simons term, viz

$\displaystyle S=N\int_{0}^{\beta}dtTr\bigg{[}\frac{1}{2}(D_{t}\Phi_{a})^{2}-\frac{1}{4}[\Phi_{a},\Phi_{b}]^{2}+\frac{2i\alpha}{3}\epsilon_{abc}\Phi_{a}\Phi_{b}\Phi_{c}\bigg{]}.$

(3.1)

This model was studied at length in [24]. The basic order parameters are as before the Polyakov line $P$ (or the corresponding eigenvalue distribution $\rho(\theta)$ of the holonomy angles) and the radius $R^{2}$. But we can also now measure the Chern-Simons term. They are given respectively by

$\displaystyle\langle|P|\rangle=\int d\theta\rho(\theta)\exp(i\theta).)$

(3.2)

$\displaystyle R^{2}=\frac{1}{\Lambda N}\langle\sum_{n=1}^{\Lambda}{Tr}{\Phi}_{i}^{2}(n)\rangle.$

(3.3)

$\displaystyle{\rm CS}=\frac{1}{N\Lambda}\langle\frac{2i}{3}\epsilon_{abc}\sum_{n=1}^{\Lambda}Tr\Phi_{a}(n)\Phi_{b}(n)\Phi_{c}(n)\rangle.$

(3.4)

It is well known that the Chern-Simons term is responsible for the emergence or condensation of the classical geometry of a round sphere (here a round sphere $\times$ Euclidean time) [18, 44, 45]. The global minimum of the theory is a fuzzy sphere given by the spin $j=(N-1)/2$ irreducible representation of $SU(2)$, i.e.

$\displaystyle\Phi_{a}=\alpha L_{a}\leavevmode\nobreak\ ,\leavevmode\nobreak\ [L_{a},L_{b}]=i\epsilon_{abc}L_{c}\leavevmode\nobreak\ ,\leavevmode\nobreak\ c_{2}=L_{a}L_{a}=\frac{N^{2}-1}{4}.$

(3.5)

Expanding around this minimum by writing $\Phi_{a}=\alpha(L_{a}+A_{a})$ leads to a gauge theory on the fuzzy sphere with a three-component gauge field $A_{a}$, i.e. we have effectively a two-dimensional gauge field tangent to the sphere coupled to a normal adjoint scalar field $\phi$ given by

$\displaystyle\phi=\frac{\Phi_{a}^{2}-c_{2}\alpha^{2}}{2\alpha^{2}\sqrt{c_{2}}}.$

(3.6)

This vacuum state is stable for large values of $\tilde{\alpha}$ defined by $\tilde{\alpha}=\alpha N^{1/3}$ but it decays (the sphere evaporates) as we decrease $\tilde{\alpha}$ to enter the Yang-Mills phase where the vacuum state becomes given by commuting matrices.

The large $d$ approximation (assuming that $D=3$ is quite large and $\alpha\longrightarrow 0$) gives immediately the action

$\displaystyle S_{0}=N\int_{0}^{\beta}dtTr\bigg{[}\frac{1}{2}(D_{t}\Phi_{a})^{2}+\frac{1}{2}m^{2}\Phi_{a}^{2}+\frac{2i\alpha}{3}\epsilon_{abc}\Phi_{a}\Phi_{b}\Phi_{c}\bigg{]}\leavevmode\nobreak\ ,\leavevmode\nobreak\ m=d^{1/3}.$

(3.7)

This action is the analogue of the cubic potential $TrM^{2}+g_{3}TrM^{3}$ which although is not stable (not bounded from below) admits a well defined large $N$ expansion. In order to access (in Monte Carlo simulations) the non-perturbative physics of this cubic matrix model we can regularize it (embed it) in a stable quartic potential of the form $TrM^{2}+g_{3}TrM^{3}+g_{4}TrM^{4}$. Indeed, it is well known that the Liouville theory (the $(2,3)-$minimal conformal matter coupled to two-dimensional quantum gravity) associated with the cubic matrix model is fully/correctly captured by the renormalization group fixed point $(g_{3*},g_{4*})=(1/432^{1/4},0)$ of the cubic-quartic matrix model [46].

Similarly here, in order to access in Monte Carlo simulations the non-perturbative physics of the matrix quantum mechanics $S_{0}$ we should regularize it by adding a quartic potential. By recalling the tracelessness condition of the original model $Tr\Phi_{a}=0$ and by demanding rotational invariance on the sphere (any single-trace term is a scalar) we conclude that there are only two possibilities we can contemplate: the single-trace term $Tr(\Phi_{a}^{2})^{2}$ and the double-trace term $(Tr\Phi_{a}^{2})^{2}$. [For example the term $Tr\Phi_{a}Tr\Phi_{a}\Phi_{b}^{2}$ is disallowed because of the tracelessness condition whereas the term $Tr\Phi_{a}\Phi_{b}Tr\Phi_{a}\Phi_{b}$ is disallowed because the individual traces are not rotational scalars].

This is very reminiscent of the multi-trace expansion of non-commutative scalar field theory [47] where the kinetic term of scalar fields on non-commutative fuzzy spaces is approximated by an expansion (hopping-parameter-like expansion) of the form (with $\Phi=U\Lambda U^{\dagger}$)

	$\displaystyle\int dU\exp(Tr[L_{a},\Phi]^{2})$	$\displaystyle=$	$\displaystyle\exp\bigg{(}a_{1}Tr\Lambda^{2}+a_{2}(Tr\Lambda)^{2}+b_{1}(Tr\Lambda^{2})^{2}+b_{2}Tr\Lambda^{2}(Tr\Lambda)^{2}+b_{3}(Tr\Lambda)^{4}$		(3.8)
		$\displaystyle+$	$\displaystyle b_{4}Tr\Lambda Tr\Lambda^{3}+b_{5}Tr\Lambda^{4}+...\bigg{)}.$

Indeed, the Yang-Mills term is essentially the kinetic term here and the analogy seems to be more than just an analogy.

The addition of the single-trace term $Tr(\Phi_{a}^{2})^{2}$ will lead to the decoupling of the normal scalar field [43] whereas the addition of the double-trace term $(Tr\Phi_{a}^{2})^{2}$ will only lead to the decoupling of the zero mode of the normal scalar field. Hence, for simplicity we are going to consider the action

$\displaystyle S_{M}=M\int_{0}^{\beta}dt(Tr\Phi_{a}^{2}-\varphi c_{2}N)^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ {c}_{2}=L_{a}L_{a}=\frac{N^{2}-1}{4}.$

(3.9)

As it turns out, this is very sufficient to regularize the model $S_{0}$ without essentially altering it. Indeed, in the large $M$ limit we obtain the condition

$\displaystyle Tr\Phi_{a}^{2}=\varphi c_{2}N.$

(3.10)

As can be checked quite easily this condition (which replaces the tracelessness condition) kills off only the zero mode of the normal scalar field $\phi$. This can be seen by substituting (3.6) in (3.9) with $\varphi=\alpha$ and assuming a fuzzy sphere background to obtain

$\displaystyle S_{M}=4\alpha^{4}N^{2}c_{2}M\int_{0}^{\beta}dt\bigg{(}\int_{{\bf S}^{2}}d\Omega\phi\bigg{)}^{2}.$

(3.11)

The mass of the zero mode of the normal scalar field is seen to be given by $m^{2}=8\alpha^{4}N^{2}c_{2}M$ which can become small at small values of the gauge coupling constant $\alpha$. But the fuzzy sphere background is typically absent in that regime. More importantly, this mass becomes suppressed at high temperatures as $m^{2}\beta=m^{2}/T$ which could be problematic (see below). This also suggests that the scaling of the mass $M$ with $N$ at high temperatures (taking also into account the scaling of the parameters $\alpha$ and $T$ given in equation (3.23) ) is given by

$\displaystyle M=\frac{M_{0}}{N^{4}}.$

(3.12)

By varying the regularized action $S_{0}+S_{M}$ with respect to $\Phi_{a}$ we obtain the classical equation of motion

$\displaystyle D_{t}^{2}\Phi_{a}=m^{2}\Phi_{a}+i\alpha\epsilon_{abc}[\Phi_{b},\Phi_{c}]+\frac{4M}{N}\Phi_{a}Tr(\Phi_{b}^{2}-\varphi c_{2}).$

(3.13)

The $SU(2)-$like configurations $\Phi_{a}=\mu J_{a}\otimes{\bf 1}_{N/n}$, where $J_{a}$ is the spin $j=(n-1)/2$ irreducible representation of $SU(2)$, are solution if the scale $\varphi$ is chosen such that

$\displaystyle\varphi=\frac{\hat{c}_{2}}{c_{2}}\mu^{2}+\frac{1}{2Mc_{2}}(\frac{m^{2}}{2}-\alpha\mu)\leavevmode\nobreak\ ,\leavevmode\nobreak\ \hat{c}_{2}=J_{a}J_{a}=\frac{n^{2}-1}{4}.$

(3.14)

The original model (with the full Yang-Mills term) admits the solutions $\mu=\alpha$ with $n=N$ being the global minimum. We choose then $\varphi$ such that $\mu=\alpha$ and $n=N$, i.e.

$\displaystyle\varphi=\alpha^{2}+\frac{1}{2Mc_{2}}\big{(}\frac{m^{2}}{2}-\alpha^{2}\big{)}.$

(3.15)

In the large $M$ limit (the limit in which the gauge field on the emergent sphere becomes truly tangential) we see that $\varphi\longrightarrow\alpha^{2}$. By substituting this expression in the previous equation we obtain the solution

$\displaystyle\mu=\frac{\alpha}{4M\hat{c}_{2}}\bigg{[}1\pm\sqrt{1+8M\hat{c}_{2}(2Mc_{2}-1)}\bigg{]}\longrightarrow\pm\frac{\alpha N}{2\sqrt{\hat{c_{2}}}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ M\longrightarrow\infty.$

(3.16)

For example, the fundamental configuration $\Phi_{a}=\mu\sigma_{a}\otimes{\bf 1}_{N/2}/2$ corresponds to a solution $\mu$ given explicitly by

$\displaystyle\mu=\frac{\alpha}{3M}\bigg{[}1\pm\sqrt{1+6M(2Mc_{2}-1)}\bigg{]}\longrightarrow\pm\frac{\alpha N}{\sqrt{3}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ M\longrightarrow\infty.$

(3.17)

This was found to be the global minimum of a similar model (Chern-Simons term+ a single-trace quartic potential in $D=0$) in [48]. In the current matrix quantum mechanics the actual global minimum (in the geometric-like phase) is a combination of $\Phi_{a}=0$ (commuting matrices) and the fundamental configuration $\Phi_{a}=\mu J_{a}$ with $J_{a}=\sigma_{a}/2$ which is given explicitly by the following embedding

$\displaystyle\boxed{\Phi_{a}=\bigg{(}\begin{array}[]{cc}\mu(J_{a})_{2\times 2}&0\\ 0&0\end{array}\bigg{)}.}$

(3.20)

This geometric-like phase is actually a three-cut phase where the three cuts correspond to the two large eigenvalues $\pm\mu/2$ (with multiplicity $1$ each) and to the small eigenvalues around $\mu=0$. This is a fuzzy-sphere-like phase because it corresponds in some sense to a fuzzy sphere with three qubits only. As we lower the coupling a transition will eventually occur to a one-cut phase (the weak coupling regime of the so-called Yang-Mills phase) where all eigenvalues are centered around the eigenvalue $\mu=0$.

This three-cut phase is termed in this article a ”baby fuzzy sphere phase” because unlike the Yang-Mills phase it is a geometric phase with all the characteristics of the fuzzy sphere phase observed in the original exact model but it comes only with three cuts or qubits, i.e. the direct sum of the irreducible representations $0$ and $1/2$ of $SU(2)$. A sample of this three-cut phase or baby fuzzy sphere phase is shown in figure (4) for temperature $\tilde{T}=0.1$ and value of the gauge coupling constant $\tilde{\alpha}=1$²²2In this section and the next we use the value $\tilde{T}=0.1$. We run the Metropolis algorithm for $2^{13}+2^{13}$ steps where the first $2^{13}$ steps are discarded whereas histograms and averages are computed using the last $2^{13}$ steps. We choose one lattice $a=0.05$ corresponding to $\Lambda=35$.. The limit $M\longrightarrow\infty$ is established quite simply by considering values of $M$ $\geq 100$.

It is not difficult to check that the configuration (3.20) is a solution of the equation of motion (3.13) with the radius $\mu$ given explicitly by

$\displaystyle\mu=\frac{\alpha}{4M\hat{c}_{2}^{\prime}}\bigg{[}1\pm\sqrt{1+8M\hat{c}_{2}^{\prime}(2Mc_{2}-1)}\bigg{]}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \hat{c}_{2}^{\prime}=\frac{2}{N}\hat{c}_{2}.$

(3.21)

The values of the radius $R^{2}$ and of the Chern-Simons term ${\rm CS}$ in this configuration (in the large $M$ limit) are given by

$\displaystyle R^{2}=\frac{\tilde{\alpha}^{2}}{4}N^{4/3}\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\rm CS}=-\frac{\tilde{\alpha}^{3}}{6^{3/2}}N^{5/2}.$

(3.22)

The scaling of the gauge coupling constant $\alpha$ and of the temperature $T$ in the large $D$ model $S_{0}+S_{M}$ is assumed to be identical to the scaling found in the exact model (with the full Yang-Mills term instead of the harmonic oscillator term with $m=d^{1/3}$), viz [24]

$\displaystyle\tilde{\alpha}=\alpha N^{1/3}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \tilde{T}=\frac{T}{N^{2/3}}.$

(3.23)

The eigenvalue distributions of the matrices $D_{a}=\Phi_{a}(n)$ (for some fixed lattice site $n$) is measured in Monte Carlo simulations by the formula

$\displaystyle\rho(\lambda)$

$\displaystyle=$

$\displaystyle\frac{1}{N}\sum_{a=1}^{N}\langle\delta(\lambda-\lambda_{a})\rangle.$

(3.24)

But from equations (3.20) and (3.21) we expect a continuum of eigenvalues centered around $\lambda=0$ together with two cuts at large values of $\lambda$ given explicitly by

	$\displaystyle\lambda_{\pm}$	$\displaystyle=$	$\displaystyle\frac{\tilde{\alpha}}{2}N^{7/6}\frac{j_{3}}{\sqrt{2\hat{c_{2}}}}$		(3.25)
		$\displaystyle=$	$\displaystyle\pm\frac{\tilde{\alpha}}{2\sqrt{6}}N^{7/6}.$

The scaling of the radius $R^{2}$, the Chern-Simons term ${\rm CS}$ and of the eigenvalues $\lambda_{\pm}$ are then given by $N^{4/3}$, $N^{5/2}$ and $N^{7/6}$ respectively which is confirmed by Monte Carlo simulations. For example, this effect is shown for the temperature $\tilde{T}=0.1$ on figures (4) (for the eigenvalues $\lambda_{\pm}$) and (5) (for $R^{2}$ and ${\rm CS}$). We also include in figure (5) the result of the Polyakov line which shows that the baby fuzzy sphere exists in the deconfined phase ($\langle|P|\rangle$ is around $1$).

Indeed, it is seen from the Monte Carlo results of the eigenvalue distribution $\rho(\lambda)$ of the matrices $D_{a}$ that the transition from the fuzzy-sphere-like phase (the three-cut phase) to the Yang-Mills phase (the one-cut phase) at (say) the fixed temperature $\tilde{T}=0.1$ occurs at a value $\tilde{\alpha}_{c}=0.625\pm 0.125$ between $1$ and $0.5$. At this critical value $\tilde{\alpha}_{c}$ the Polyakov line switches from $0$ in the Yang-Mills phase to $1$ in the fuzzy-sphere-like phase. On the other hand, the Chern-Simons term vanishes whereas the radius becomes very small in the Yang-Mills phase. The critical line in the plane $\tilde{\alpha}-\tilde{T}$ can be constructed along these lines, i.e. by following the behavior of the eigenvalue distribution $\rho(\lambda)$ which we will do shortly.

The small eigenvalues in (3.20) (which are fluctuating around $0$ with a degeneracy equal $N-2$) will play a greater role in the Yang-Mills phase (which is the phase encountered for smaller values of $\tilde{\alpha}$) and thus their behavior will be discussed next.

The eigenvalue distribution $\rho(\lambda)$ in the Yang-Mills phase becomes effectively a Wigner semi-circle law and it can be computed starting from the following two assumptions (confirmed by Monte Carlo simulations). First, the Chern-Simons term vanishes in the Yang-Mills phase indicating the dominance of commuting and diagonal matrices. Second, the three commuting matrices $D_{a}$ are static and by rotational invariance their contributions are identical. The matrix model $S_{0}+S_{M}$ leads then to the effective potential (with $\beta_{1}=\beta.d$ and $\beta_{2}=\beta.d^{2}$)

$\displaystyle V=N\beta_{1}(\frac{m^{2}}{2}-2Mc_{2}\varphi)TrM^{2}+M\beta_{2}(TrM^{2})^{2}.$

(3.26)

The saddle point equation of this matrix model is identical to the saddle point equation of the quadratic matrix model $BTrM^{2}$ with the mass $B$ given by

$\displaystyle B=N\beta_{1}(\frac{m^{2}}{2}-2Mc_{2}\varphi)+2M\beta_{2}Na_{2}=NB_{0}+NB_{1}a_{2}.$

(3.27)

The difference with the usual quadratic matrix model is the fact that the parameter $B$ depends on the second moment $a_{2}=TrM^{2}/N$. The solution of the matrix model $BTrM^{2}$ is known to be given by the Wigner semi-circle law

$\displaystyle\rho(\lambda)=\frac{2}{\pi a^{2}}\sqrt{a^{2}-\lambda^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ a^{2}=\frac{2N}{B}.$

(3.28)

From this solution we compute the second moment $a_{2}=\int_{-a}^{+a}d\lambda\rho(\lambda)\lambda^{2}=a^{2}/4=N/2B$. This leads to the consistency condition $Ba_{2}=N/2$ or equivalently to the quadratic equation $B_{1}a_{2}^{2}+B_{0}a_{2}-1/2=0$ which admits the solution

$\displaystyle a_{2}=\frac{\sqrt{B_{0}^{2}+2B_{1}}-B_{0}}{2B_{1}}.$

(3.29)

By substituting in the mass parameter $B$ we obtain the solution

$\displaystyle B=\frac{N}{2}(\sqrt{B_{0}^{2}+2B_{1}}+B_{0}).$

(3.30)

Hence, the radius of the Wigner semi-circle is given by

$\displaystyle a^{2}=2\frac{\sqrt{B_{0}^{2}+2B_{1}}-B_{0}}{B_{1}}.$

(3.31)

Here there are two quite different behaviors. First, for $\tilde{\alpha}\neq 0$ we get for large $M$ the solution (corresponding to commuting matrices)

$\displaystyle a^{2}=\frac{4\beta_{1}c_{2}\varphi}{\beta_{2}}+\frac{2}{B_{1}}(\frac{\beta_{2}}{\beta_{1}c_{2}\varphi}-\beta_{1}m^{2})+O(\frac{1}{M^{2}})\Rightarrow a=\frac{\tilde{\alpha}N^{2/3}}{\sqrt{d}}+O(\frac{1}{M}).$

(3.32)

This leading value of the radius is independent of $M$ and $\tilde{T}$, it depends linearly on $\tilde{\alpha}$ and scales slowly with $N$ as $N^{2/3}$.

However, for $\tilde{\alpha}=0$ the behavior for large $M$ is found to be distinctly different (corresponding to the minimum $\Phi_{a}=0$ exactly) given by the law

$\displaystyle a=(\frac{4}{M\beta d^{2}})^{1/4}.$

(3.33)

Both behaviors (3.32) and (3.33) are confirmed in Monte Carlo simulations. See the two graphs on figure (6) for $\tilde{\alpha}=0.2$ and $\tilde{\alpha}=0.1$ respectively (with $\tilde{T}=0.1$ and $N=12$).

For later purposes, it is also important to keep track of the leading $1/M$ correction which contains the $T-$dependence of the result (3.32). We have then the radius of the Wigner semi-circle law

$\displaystyle a^{2}=\frac{4c_{2}\alpha^{2}}{d}+\frac{1}{Md}(\frac{T}{c_{2}\alpha^{2}}-2\alpha^{2})+O(\frac{1}{M^{2}}).$

(3.34)

This behavior is also confirmed in Monte Carlo simulations for high temperatures (see next section).

The radius $R^{2}$ can then be calculated by the formula

	$\displaystyle R^{2}$	$\displaystyle=$	$\displaystyle da_{2}$		(3.35)
		$\displaystyle=$	$\displaystyle\frac{d}{4}a^{2}$
		$\displaystyle=$	$\displaystyle c_{2}\alpha^{2}+\frac{1}{4M}(\frac{T}{c_{2}\alpha^{2}}-2\alpha^{2})+O(\frac{1}{M^{2}}).$

Thus, the leading behavior agrees with the result in the baby sphere phase given by the first equation in (3.22) which indicates that this observable is continuous across the critical point. Hence, the leading $1/M$ correction must also be continuous.

By approximating $Tr\Phi_{a}^{2}/N$ by $R^{2}$ and then using equation (3.35) we can rewrite the equation of motion (3.13) for static configurations as

	$\displaystyle-i\alpha\epsilon_{abc}[\Phi_{b},\Phi_{c}]$	$\displaystyle=$	$\displaystyle 2(\alpha^{2}+2MR^{2}-2Mc_{2}\alpha^{2})\Phi_{a}$		(3.36)
		$\displaystyle=$	$\displaystyle\frac{T}{c_{2}\alpha^{2}}\Phi_{a}.$

This lead to the fuzzy sphere algebra given by

$\displaystyle[\Phi_{a},\Phi_{b}]=\frac{i}{N^{1/3}}\frac{2\tilde{T}}{\tilde{\alpha}^{3}}\epsilon_{abc}\Phi_{c}.$

(3.37)

Naturally this is vanishingly small in the large $N$ limit which shows explicitly how the geometry decays in the Yang-Mills phase.