## LATEX

### Matrix models and noncommutative gravity

Yang-Mills matrix models

Yang-Mills matrix models play a crucial role in noncommutative gravity and emergent geometry. As an example we will consider noncommutative ${\bf AdS}^2_{\theta}$ which can be obtained as the classical background solution of the following $D=3$  matrix model
\begin{eqnarray}
\end{eqnarray}
The ambient metric is $\eta=(-1,+1,+1)$, $D_a=(D_a)^{\dagger}$ are three matrices in ${\rm Mat}(\infty,\mathbb{C})$ and $f_{abc}$ are the structure constants of $SO(1,2)$. Hence this model is invariant under $SO(1,2)$ rotations as well as under gauge transformations $X_a\longrightarrow UX_aU^{\dagger}$ and under  translations $X_a\longrightarrow X_a+c$.

The variation of the action and the equation of motion are given by
\begin{eqnarray}
\delta S=-Tr\delta D^a[D^b,F_{ab}]\equiv 0\Rightarrow F_{ab}=[D_a,D_b]-i\kappa f_{abc}D^c\equiv 0.\label{eom}
\end{eqnarray}
A solution of these equations of motion is given by
\begin{eqnarray}
\end{eqnarray}
The $J^a$ are the generators of the Lie group $SO(1,2)$ in the irreducible representation given by the discrete series $D_j^{\pm}$ which are labeledby an integer $j\gt 1$. The $\hat{X}^a$ are thus precisely the coordinate operators on noncommutative ${\bf AdS}^2_{\theta}$ and as a consequence we have
\begin{eqnarray}
\end{eqnarray}
In other words, the radius $R$ of  noncommutative ${\bf AdS}^2_{\theta}$, the integer $j$ labeling the discrete series $D_j^{\pm}$ and the coupling constant $\kappa$ of the corresponding Yang-Mills matrix model (\ref{YM}) are related by the condition

\begin{eqnarray}
R^2=\kappa^2j(j-1).
\end{eqnarray}
Therefore, the commutative limit $\kappa\longrightarrow 0$ corresponds to the large representation limit  $j\longrightarrow \infty$. The geometric commutative limit can be thought of as the semi-classical limit.

In noncommutative gravity the fundamental degrees of freedom of the theory are given by the hermitian  matrices $D^a$ and not by the metric $g_{\mu\nu}$ which can only emerge in the semi-classical/commutative limit $\kappa\longrightarrow 0$ (or equivalently $j\longrightarrow \infty$) as outlined in \cite{Steinacker:2008ri,Steinacker:2010rh}.

Furthermore, the noncommutativity tensor $\theta^{ab}$ (or equivalently the Poisson structure  $\theta^{\mu\nu}$ of the underlying symplectic manifold ${\bf AdS}^2$) is generically a function of the matrices $D^a$ (or equivalently of the local coordinates $x^{\mu}$ on ${\bf AdS}^2$) and plays also a more fundamental role than the emergent metric $g_{\mu\nu}$. This tensor is given explicitly by
\begin{eqnarray}
[D^a,D^b]=i\kappa\theta^{ab}(D).\label{commutator}
\end{eqnarray}
Clearly, in the classical (classical with respect to the matrix model) configurations $D^a\equiv \hat{X}^a$ we must have $\theta^{ab}(D)\equiv f^{abc}\hat{X}_c$.

The matrix coordinates $D_a=\hat{X}^a$ behave in the commutative limit as $D_a \sim X^a$ which are the embedding coordinates of ${\bf AdS}^2$. These coordinates can always be decomposed into tangential and normal coordinates on ${\bf AdS}^2$. This can be seen  by considering for example the neighborhood of the "north pole", viz $X^3\equiv\phi\simeq R$, $X^{1}\equiv x^1 \lt\lt R$ and $X^2\equiv x^2\lt\lt R$ where $x^{1}$ and $x^{2}$ are local coordinates on ${\bf AdS}^2$. The commutator (\ref{commutator}) around the "north pole" becomes then $[\hat{x}^{\mu},\hat{x}^{\nu}]=i\kappa \theta^{\mu\nu}$ where $\theta^{\mu\nu}=Rf^{\mu\nu 3}$.

We decompose then the matrices $D^a\equiv \hat{X}^a$ into tangential and normal components as
\begin{eqnarray}
\hat{X}^a=(\hat{x}^{\mu},\hat{\phi}).
\end{eqnarray}
From the requirement (\ref{casimir}) we can see that $\phi$ is a function of $\hat{x}^{\mu}$, $\mu=1,2$, i.e.

\begin{eqnarray}
\hat{\phi}=\hat{\phi}(\hat{x}).
\end{eqnarray}
Hence, the commutator (\ref{commutator}) becomes
\begin{eqnarray}
[\hat{x}^{\mu},\hat{x}^{\nu}]=i\kappa\theta^{\mu\nu}(\hat{x})~,~\theta^{\mu\nu}\equiv f^{\mu\nu3}\hat{\phi}(\hat{x}).
\end{eqnarray}
The quantized derivations (parallel and normal) on noncommutative ${\bf AdS}^2_{\theta}$ and their commutative counterparts  on  ${\bf AdS}^2$ are then given by
\begin{eqnarray}
\hat{e}^a(F)=-i[D^a,F]\longrightarrow e^a(F)=\kappa\theta^{\mu\nu}\partial_{\mu}x^a\partial_bF.
\end{eqnarray}
Next, we introduce the covariant form of the action (\ref{action}) by
\begin{eqnarray}
S[D,\hat{\Phi}]=\frac{2\pi R \kappa}{2}Tr\bigg(-\frac{1}{R^2\kappa^2}[D^a,\hat{\Phi}][D_a,\hat{\Phi}]+m^2\hat{\Phi}^2\bigg).\label{actioncov}
\end{eqnarray}
We can now compute in the configurations $D_a=\hat{X}^a$ the kinetic term

\begin{eqnarray}
-\eta_{ab}[D^{a},\hat{\Phi}][D^{b},\hat{\Phi}]&=&\eta_{ab}\hat{e}^{a}(\hat{\Phi})\hat{e}^{b}(\hat{\Phi})\nonumber\\
&\sim &\kappa^{2}\theta^{\mu\mu^{\prime}}\theta^{\nu\nu^{\prime}}{g}_{\mu\nu}{\partial}_{\mu^{\prime}}\Phi{\partial}_{\nu^{\prime}}\Phi\nonumber\\
&\sim &{G}^{\mu^{\prime}\nu^{\prime}}{\partial}_{\mu^{\prime}}\Phi{\partial}_{\nu^{\prime}}\Phi.
\end{eqnarray}
The quantity $G^{\mu\nu}$ is the induced metric which couples to matter fields and which is given explicitly by
\begin{eqnarray}
{G}^{\mu^{\prime}\nu^{\prime}} &=&\kappa^{2}\theta^{\mu\mu^{\prime}}\theta^{\nu\nu^{\prime}}{g}_{\mu\nu}.
\end{eqnarray}
Whereas $g_{\mu\nu}$ is  the embedding metric (the metric on ${\bf AdS}^2$ viewed as a Poisson manifold) given explicitly by
\begin{eqnarray}
{g}_{\mu\nu}&=& \eta_{ab}{\partial}_{\mu}x^{a}{\partial}_{\nu}x^{b}.
\end{eqnarray}
The kinetic action is then given by
\begin{eqnarray}
-Tr [D_{a},\hat{\Phi}][D^{b},\hat{\Phi}]
&\sim &\frac{1}{2\pi}\int d^2x\rho(x){G}^{\mu^{\prime}\nu^{\prime}}{\partial}_{\mu^{\prime}}\Phi{\partial}_{\nu^{\prime}}\Phi.\label{kaction}
\end{eqnarray}
We introduced in this last equation a scalar density $\rho$, which  defines on the quantized  Poisson manifold ${\bf AdS}^2_{\theta}$ a local non-commutativity scale, by the relation
\begin{eqnarray}
\rho=\frac{1}{\sqrt{{\rm det}{\kappa \theta^{\mu\nu}}}}.\label{sd1}
\end{eqnarray}
The kinetic action (\ref{kaction}) does not have the canonical covariant form which can be reinstated by a rescaling of the metric as follows

\begin{eqnarray}
\tilde{G}^{ab}=\exp(-\sigma)G^{ab}.\label{sd2}
\end{eqnarray}
And imposing the condition
\begin{eqnarray}
\rho G^{ab}=\sqrt{{\rm det}\tilde{G}_{ab}}\tilde{G}^{ab}\Rightarrow \rho=\sqrt{{\rm det}G_{ab}}.\label{sd3}
\end{eqnarray}
By using equations  (\ref{sd1}) and (\ref{sd3}) we can show that the scalar density $\rho$ can also be written in the form $\rho=\sqrt{{\rm det}g_{ab}}$. Hence, we must have
\begin{eqnarray}
G_{ab}\equiv g_{ab}.
\end{eqnarray}
And by substituting in (\ref{sd2}) we obtain
\begin{eqnarray}
\tilde{G}^{ab}\equiv e^{-\sigma} g^{ab}.
\end{eqnarray}We get immediately in the semi-classical limit $\kappa\longrightarrow 0$ the kinetic action
\begin{eqnarray}
-Tr [D_{a},\hat{\Phi}][D^{b},\hat{\Phi}]
&\sim &\frac{1}{2\pi}\int d^2x\sqrt{{\rm det}G_{\mu\nu}}{G}^{\mu^{\prime}\nu^{\prime}}{\partial}_{\mu^{\prime}}\Phi{\partial}_{\nu^{\prime}}\Phi\nonumber\\
&\sim & \frac{1}{2\pi}\int d^2x\sqrt{{\rm det}\tilde{G}_{\mu\nu}}{\tilde{G}}^{\mu^{\prime}\nu^{\prime}}{\partial}_{\mu^{\prime}}\Phi{\partial}_{\nu^{\prime}}\Phi.
\end{eqnarray}
The conformal factor $e^{-\sigma}$ remains therefore undetermined since in two dimensions Weyl transformations of the metric $G^{\mu\nu}\longrightarrow e^{-\alpha}G^{\mu\nu}$ are  in fact symmetries of the action \cite{Jurman:2013ota}.

By going through the same steps we can now show that the Yang-Mills term (quartic term) of the matrix model (\ref{YM}) reduces, in the semi-classical/commutative limit $\kappa\longrightarrow 0$ (or equivalently $j\longrightarrow \infty$), not to the Einstein equations but to the cosmological term \cite{Jurman:2013ota}. A matrix form of the Einstein equations can also be written down but this is not necessary within the formalism of noncommutative gravity since the condensation of the geometry of ${\bf AdS}^2_{\theta}$ is in fact driven by the Myers-Chern-Simons term (cubic term)  of  (\ref{YM}) \cite{Myers:1999ps}.

Indeed, the ${\bf AdS}^2_{\theta}$ solution (\ref{ads2}) of the equation of motion (\ref{eom}) is not unique and this solution can be made more stable by adding a potential term the action (\ref{YM}) which implements explicitly the constraint (\ref{casimir}) such as the term \cite{CastroVillarreal:2004vh}
\begin{eqnarray}
\end{eqnarray}
The action (\ref{YM})+(\ref{potential}) will then admit for large and positive values of $M^2$ a unique solution given by the ${\bf AdS}^2_{\theta}$ background (\ref{ads2}) which satisfies the constraint (\ref{casimir}) by construction. The expansion of the scalar action (\ref{actioncov})  around the AdS solution becomes more reliable since this background in the limit $M^2\longrightarrow \infty$ is  completely stable. Therefore, the action (\ref{YM})+(\ref{potential}) acts effectively within noncommutative gravity as an Einstein-Hilbert action.

References

%\cite{Steinacker:2008ri}
\bibitem{Steinacker:2008ri}
H.~Steinacker,
Emergent Gravity and Noncommutative Branes from Yang-Mills Matrix Models,''
Nucl.\ Phys.\ B {\bf 810}, 1 (2009)
%doi:10.1016/j.nuclphysb.2008.10.014
[arXiv:0806.2032 [hep-th]].
%%CITATION = doi:10.1016/j.nuclphysb.2008.10.014;%%
%74 citations counted in INSPIRE as of 22 Mar 2019

%\cite{Steinacker:2010rh}
\bibitem{Steinacker:2010rh}
H.~Steinacker,
Emergent Geometry and Gravity from Matrix Models: an Introduction,''
Class.\ Quant.\ Grav.\  {\bf 27}, 133001 (2010)
%  doi:10.1088/0264-9381/27/13/133001
[arXiv:1003.4134 [hep-th]].
%%CITATION = doi:10.1088/0264-9381/27/13/133001;%%
%138 citations counted in INSPIRE as of 27 Mar 2019

%\cite{Myers:1999ps}
\bibitem{Myers:1999ps}
R.~C.~Myers,
Dielectric branes,''
JHEP {\bf 9912}, 022 (1999)
doi:10.1088/1126-6708/1999/12/022
[hep-th/9910053].
%%CITATION = doi:10.1088/1126-6708/1999/12/022;%%
%1285 citations counted in INSPIRE as of 16 Nov 2019

%\cite{CastroVillarreal:2004vh}\bibitem{CastroVillarreal:2004vh}
A Gauge-invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere,''
  %%CITATION = doi:10.1016/j.nuclphysb.2004.10.032;%%
%54 citations counted in INSPIRE as of 16 Nov 2019