## LATEX

### The multitrace matrix models of noncommutative geometry and quantum gravity

In here we are interested in a noncommutative scalar $\Phi^4$ theory. As an example, we will take the underlying noncommutative space to be fuzzy ${\bf CP}^n_l$ which is obtained by Berezin quantization \cite{Berezin:1974du} of the projective space ${\bf CP}^n=SU(n+1)/U(n)$.

Fuzzy  ${\bf CP}^n_l$ is given by the spectral triple \cite{Frohlich:1993es,Connes:1994yd} (see also \cite{Dolan:2001gn})
\begin{eqnarray}
{\bf CP}^n_l=({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l}).
\end{eqnarray}
${\bf H}_{n,l}$ is the Hilbert space associated with the irreducible representation of $su(n+1)$ which is given by the totally symmetrized tensor product of  $l$ fundamental representations, viz $(l,0,0,...,0)$. The dimension of this representation is given by
\begin{eqnarray}
N_{n,l}=\frac{(n+l)!}{n!l!}.
\end{eqnarray}
The Hilbert space ${\bf H}_{n,l}$ is thus acted on by the complete matrix algebra ${\rm Mat}_{N_{n,l}}$ of finite dimension $N_{n,l}$ with inner product defined by
\begin{eqnarray}
(f,g)=\frac{1}{N_{n,l}}Tr f^{\dagger}g.
\end{eqnarray}
The $\Delta_{n,l}$ is the Laplace operator on fuzzy ${\bf CP}^n_l$ defined, in terms of the generators $L_i$ of $su(n+1)$ in the representation $(l,0,0,...,0)$, by the quadratic Casimir
\begin{eqnarray}
\Delta_{n,l}(f)=[L_i,[L_i,f]].
\end{eqnarray}
Derivation on  fuzzy ${\bf CP}^n_l$ are given precisely by the adjoint action of the generators $L_i$, i.e.
\begin{eqnarray}
{\rm ad}L_i(f)=[L_i,f]=(L_i^L-L_i^R)(f).
\end{eqnarray}
This shows explicitly that the space of functions on fuzzy ${\bf CP}^n_l$, which is the matrix algebra ${\rm Mat}_{d_{n,l}}$, decomposes under the action of $SU(n+1)$ as the direct sum of the  irreducible representations $(m,...,m)$  of $SU(n + 1)$,viz
\begin{eqnarray}
(l,...,0)\otimes \overline{(l,...,0)}=\oplus_{m=0}^l(m,0,...,0,m).
\end{eqnarray}
Hence, the matrix algebra ${\rm Mat}_{N_{n,l}}$ is the endomorphism ${\bf Hom}(V_{n,l})=V_{n,l}\otimes V_{n,l}^*$ where $V_{n,l}$ is the vector space associated with the representation $(l,0,...,0)$.

The polarization tensors $T_{m \sigma}$  which transform in the  irreducible representation $(m,...,m)$ are the eigenmatrices of the Laplace $\Delta_{n,l}$ with eigenvalue $\lambda_m$ and degeneracy $d_{n,m}$ given by
\begin{eqnarray}
\lambda_m=2m(m+n).
\end{eqnarray}
And
\begin{eqnarray}
d_{n,m}=\frac{n(2m+n)((m+n-1)!)^2}{(n!)^2(m!)^2}.
\end{eqnarray}
Indeed, the index $\sigma$ in $T_{m \sigma}$ denotes the other quantum numbers required to specify the representation $(m,...,m)$.

The above spectrum is precisely the spectrum of the Laplace operator on commutative ${\bf CP}^n$ only cutoff at $m=l$. The commutative limit is therefore $l\longrightarrow\infty$ in which the eigenmatrices $T_{m \sigma}$ go over to the correct eigenfunctions $Y_{m \sigma}$ (spherical harmonics) on  ${\bf CP}^n$.

The coordinate functions $X_i$  on fuzzy ${\bf CP}^n_l$ are obtained from suitably
rescaling the generators $L_i$ of $su(n + 1)$ in the representation $(l,0,...,0)$, namely $X_i=a_l L_i$. We have then the commutation relations
\begin{eqnarray}
[X_i,X_j]=ia_l f_{ijk}X_k.
\end{eqnarray}
The data contained in the spectral triple  $({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l})$ is sufficient to write down an Euclidean action for a scalar field theory on fuzzy  ${\bf CP}^n_l$. A scalar field $\Phi$ is a hermitian $N\times N$ matrix in ${\rm Mat}_{N_{n,l}}$, i.e. $N\equiv N_{n,l}$. An action on fuzzy  ${\bf CP}^n_l$ is then given by
\begin{eqnarray}
S=Tr(\Phi C_2\Phi+r\Phi^2+g\Phi^4).
\end{eqnarray}
The partition function of the model is given by
\begin{eqnarray}
Z=\int d\mu_D(\Phi)\exp(-\beta S[\Phi]).
\end{eqnarray}
$d\mu_D(\Phi)$ is Dyson measure on ${\rm Mat}_{N}$.

We can diagonalize the matrix $\Phi$ as $\Phi=U\Lambda U^{\dagger}$. We also write $\Phi=\Phi_{\mu}T_{\mu}$ or $\Phi_{\mu}=Tr\Phi T_{\mu}$ where $T_{\mu}$, $\mu=1,...,N^2$ are the generators of $u(N)$. The partition function then becomes

\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\
&\times &\int d\mu_H(U)\exp\big(-\beta K_{\mu\nu}Tr U\Lambda U^{\dagger}T_{\mu}Tr U\Lambda U^{\dagger}T_{\nu}\big).
\end{eqnarray}
The Vandermonde $\Delta^2(\Lambda)$ is given by
\begin{eqnarray}
\Delta^2(\Lambda)=\prod_{i=1}^N(\lambda_i-\lambda_j)^2.
\end{eqnarray}
And $d\mu_H(U)$ is the Haar measure on $U(N)$. The kinetic matrix $K_{\mu\nu}$ is given on the other hand by
\begin{eqnarray}
K_{\mu\nu}=-Tr[L_i,T_{\mu}][L_i,T_{\nu}].
\end{eqnarray}
We can now perform a hopping parameter expansion to perform the integral over $U$ (by using the properties $(A\otimes B)(C\otimes D)=(AC\otimes BD)$ and $Tr(A\otimes B)=TrA TrB$ and the orthogonality relation of the Haar measure) as follows \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\
&\times &\sum_k\frac{(-\beta)^k}{k!}\int d\mu_H(U)\prod_{i=1}^kK_{\mu_i\nu_i}Tr U\Lambda U^{\dagger}T_{\mu_i}Tr U\Lambda U^{\dagger}T_{\nu_i}\nonumber\\
&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\
&\times & \int d\mu_H(U)Tr \bigg((U\otimes...\otimes U)(\Lambda\otimes...\otimes\Lambda)(U^{\dagger}\otimes...\otimes U^{\dagger})(T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k})\bigg)\nonumber\\
&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\
&\times &\sum_{\rho}\frac{1}{{\rm dim}(\rho)}  Tr_{\rho}(\Lambda\otimes...\otimes\Lambda)Tr (T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k}).
\end{eqnarray}
After a very long and involved calculation we find upto the order $O(\beta^2)$  the multitrace action \cite{Saemann:2010bw,OConnor:2007ibg}

\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\exp(-\beta (S_1+S_2)),
\end{eqnarray}
where
\begin{eqnarray}
S_1={\cal J}_1=a_{2,0}Tr\Lambda^2+a_{1,1}(Tr\Lambda)^2.
\end{eqnarray}
And
\begin{eqnarray}
S_2&=&\frac{\beta}{2}({\cal J}_1^2-{\cal J}_2)\nonumber\\
{\cal J}_2&=&a_{4,0}Tr\Lambda^4+a_{3,1}Tr\Lambda^3 Tr\Lambda+a_{2,2}(Tr\Lambda^2)^2+a_{2,1,1}Tr\Lambda^2(Tr\Lambda)^2+a_{1,1,1,1}(Tr\Lambda)^4.
\end{eqnarray}
The coefficients $a$ can be found in \cite{Saemann:2010bw,OConnor:2007ibg}.

For example, we compute on ${\bf CP}^1_l$, where the highest weight $l$ is related to the ordinary spin $s$ by $s=l/2$, the following quantities $N=l+1$, $Tr K\sim N^4$, $Tr K^2\sim 4N^6/3$, $Tr K^3\sim 2N^8$ and $a_{3,1}\sim 8/3$.

Whereas we compute in four dimensions on ${\bf CP}^2_l$ the following quantities $N=(l+1)(l+2)/2$, i.e. $l^2\sim 2N$ and  as a consequence $Tr K\sim 8 N^3/3$, $Tr K^2\sim 8N^4$, $Tr K^3\sim 128 N^5/5$ and $a_{3,1}\sim -256/9N^2$.

It was argued in \cite{Ydri:2017riq} that the non-vanishing of the  multitrace term proportional to $a_{3,1}$ is a necessary and a sufficient condition for a stable Ising phase and a stable emergent background geometry. It was also argued there  that quantum gravity in (or more precisely quantized spaces of)  dimensions higher than two can thus be obtained from pure multitrace matrix models along these same lines.

By neglecting all multitrace terms we find that the quartic matrix model $N Tr(\tilde{r}\Phi^2+\tilde{g}\Phi^4)$ (where $\tilde{r}=r/N^{3/2}$ and $\tilde{g}=g/N^2$) in the limit of very weak coupling $\tilde{g}\longrightarrow 0$ is dominated by the Wigner semi-circle law given by
\begin{eqnarray}
\rho(\lambda)=\frac{\tilde{r}}{\pi}\sqrt{\frac{2}{\tilde{r}}-\lambda^2}.
\end{eqnarray}
However, by including all multitrace terms we get noncommutative $\Phi^4$ on fuzzy ${\bf CP}^n_l$ which is also dominated (or more precisely its eigenvalue sector) in the limit of weak coupling $\tilde{g}\longrightarrow 0$ by a Wigner semi-circle corresponding to the following simple model \cite{Steinacker:2005wj}
\begin{eqnarray}
S=\frac{2N}{\alpha_0^2(m)}Tr\Phi^2.
\end{eqnarray}
In other words, we must make in the above Wigner semi-circle law the following replacement

\begin{eqnarray}
\tilde{r}\longrightarrow\frac{2}{\alpha_0^2(m)}.
\end{eqnarray}
$\alpha_0$ is the maximum eigenvalue and it is given in terms of the cutoff $\Lambda\sim l$ on fuzzy ${\bf CP}^n_l$ and the mass $m^2\sim r$ by the relation \cite{Steinacker:2005wj}
\begin{eqnarray}
\alpha_0^2(m)=4c(m,\Lambda)\Lambda^{d-2}.
\end{eqnarray}
We have explicitly
\begin{eqnarray}
c(m,\Lambda)=\frac{1}{16\pi^2}\bigg(1-\frac{m^2}{\Lambda^2}\ln(1+\frac{\Lambda^2}{m^2})\bigg)~,~d=4,n=2.
\end{eqnarray}

\begin{eqnarray}
c(m,\Lambda)=\frac{1}{4\pi}\ln(1+\frac{\Lambda^2}{m^2})~,~d=2,n=1.
\end{eqnarray}
Thus, by studying the multitrace approximation at weak coupling we can determine when the Wigner behavior goes from the behavior of the pure matrix model $d=0$ to the behavior of noncommutative matrix models at $d=2$ and $d=4$ as we vary for example the parameter $a_{3,1}$. The emergence of the correct behavior is an indication of a stable Ising phase and as a consequence a stable emergent background  geometry.

#### References

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