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} S[D]=Tr(-\frac{1}{4}[D_a,D_b][D^a,D^b]+\frac{2i}{3}\kappa f_{abc}D^aD^bD^c).\label{YM} \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} D^a=\kappa J^a\equiv \hat{X}^a.\label{ads2} \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} D^aD_a=\hat{X}^a\hat{X}_a=\kappa^2J^aJ_a=-R^2.\label{casimir} \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} V[D]=M^2Tr(D^aD_a+R^2)^2. \label{potential} \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}
P.~Castro-Villarreal, R.~Delgadillo-Blando and B.~Ydri,
``A Gauge-invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere,''
Nucl.\ Phys.\ B {\bf 704}, 111 (2005)
doi:10.1016/j.nuclphysb.2004.10.032
[hep-th/0405201].
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} S[D]=Tr(-\frac{1}{4}[D_a,D_b][D^a,D^b]+\frac{2i}{3}\kappa f_{abc}D^aD^bD^c).\label{YM} \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} D^a=\kappa J^a\equiv \hat{X}^a.\label{ads2} \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} D^aD_a=\hat{X}^a\hat{X}_a=\kappa^2J^aJ_a=-R^2.\label{casimir} \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} V[D]=M^2Tr(D^aD_a+R^2)^2. \label{potential} \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}
P.~Castro-Villarreal, R.~Delgadillo-Blando and B.~Ydri,
``A Gauge-invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere,''
Nucl.\ Phys.\ B {\bf 704}, 111 (2005)
doi:10.1016/j.nuclphysb.2004.10.032
[hep-th/0405201].
%%CITATION = doi:10.1016/j.nuclphysb.2004.10.032;%% %54 citations counted in INSPIRE as of 16 Nov 2019
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