\da{{\dot\alpha},\lambda} \def\L{\Lambda} \def\la{\lambda^\beta}           \def\db{{\dot\beta}]  \nonumber\\
&\stackrel{k_i \to 0}{\rightarrow} 
\cF_\alpha}          \def\da{{\dot\alpha}^{ab}(k_1) f_\beta}           \def\db{{\dot\beta}(k_2)e^{i (k_1+k_2) X}  [\lambda} \def\L{\Lambda} \def\la{\lambda^\alpha}          \def\da{{\dot\alpha},\lambda} \def\L{\Lambda} \def\la{\lambda^\beta}           \def\db{{\dot\beta}]
\,, 
\end{align}
leading to the same $\mathfrak{s}\mathfrak{u}(N)$-valued contributions as in the commutative case in the low energy limit  $k_i \to 0$, while NC effects are sub-leading. 
This means, that in the supersymmetric case one may expect the low-energy effective action to reduce to $\cN=4$ SYM on a general background ${\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}$.



In Ref.~\cite{Blaschke:2011qu}, it was shown that the effective action for an unbroken massless gauge boson in  the Coulomb phase on a generic curved brane leads to the leading terms of the Dirac-Born-Infeld (DBI) action for a single brane at some distance of the stack of $N-1$ branes: 
Consider $SU(N)$ broken down to $SU(N-1)\times U(1)$ through scalar fields $\phi^i\sim\lambda$ where $\lambda$ is the generator of the unbroken $U(1)$.
Then the one-loop effective action agrees with expansion of the Dirac-Born-Infeld (DBI) action for a D3-brane  in the background of $N-1$ coinciding branes\footnote{For the detailed computations in this so-called Coulomb branch, we once more refer to Ref.~\cite{Blaschke:2011qu}.}
\begin{align}
 S_{\rm DBI} &=
T_3\! \int_\mathcal{M} \!d^4 x\, \frac{|\phi^2|^2}{Q}\Bigg(\sqrt{\bigg|\det\!\Big(G_{\mu\nu} + \tfrac{Q}{|\phi^2|^2}D_\mu\phi^i D_\nu\phi_i
+ \tfrac{Q^{1/2}}{|\phi^2|} \frac{F_{\mu\nu}}{\L_{\rm NC}^{2}}\Big)\bigg|} -\sqrt{|\det G|}\Bigg), \nonumber \\
 Q&=\frac{(N-1)}{2\pi^2\L_{\rm NC}^4}\,, 
\qquad\qquad T_3=\L_{\rm NC}^4
\,. \label{DBI-action-AdS}
\end{align}
In string theory, these constants are known to be $Q=\frac{(N-1) g_s{\alpha}          \def\da{{\dot\alpha}'}^2}{\pi}$ and $T_3 = \frac 1{2\pi g_s{\alpha}          \def\da{{\dot\alpha}'}^2}$ \cite{Tseytlin:1999dj}, which is consistent with the above result. 
In fact, it allows to identify the scale of non-commutativity with the string theory parameters as 
\begin{align}
g_s\alpha'^2=\frac1{2\pi\L_{\rm NC}^4}
\,. 
\end{align}
The geometrical meaning of \eq{DBI-action-AdS} may be understood as follows: 
A single D3 brane is modelled by the unbroken $U(1)$ component, whose displacement in the transversal $\rho^6$ is given by $\phi^i$. Hence, for small transversal distance $|\phi|$ or large $N$ and $G_{\mu\nu} = \eta_{\mu\nu}$, the  action \eq{DBI-action-AdS} reduces to the DBI action on a geometry with effective metric
\begin{align}
ds^2 &= H^{-1/2}(x) dx^\mu dx_\mu + H^{1/2}(x)(d\phi^2 + \phi^2 d\Omega^5)\,, \nonumber\\
H &= 1+ \frac{Q}{|\phi^2|^2} \;\, \approx \, \frac{Q}{|\phi^2|^2}  \qquad\mbox{for}\,\,\frac{Q}{|\phi^2|^2} \gg 1
\,. 
\end{align}
This is consistent with IIB supergravity \cite{Buchbinder:2001ui,Maldacena:1997re}, 
and as is well-known, reduces to $AdS^5 \times S^5$ in the near-horizon limit. 
The result above is in fact more general, since the effective 4-dimensional brane metric $G_{\mu\nu}$ is not necessarily flat. 
Therefore the quantum corrections to the matrix model can be interpreted in terms of a modified bulk geometry of the embedding space $\rho^{10}$. 

Note that in contrast to previous work on related issues such as~\cite{Maldacena:1999mh,Alishahiha:1999ci,Berman:2000jw,Liu:2000ad}, the results above are exclusively based on matrix model computations, without presupposing any relation with supergravity.


More general backgrounds, which in contrast to the situation above break the $SU(N)$ gauge symmetry completely, and which can be viewed either as product space ${\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}^4 \times \cK$, or in terms of a $SU(N)$ gauge theory on ${\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}^4$ in the Higgs phase, may also be considered. 
Such backgrounds will typically break supersymmetry and hence lead to induced gravity terms. 
The non-Abelian fields ${\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}^a={\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}^a_\alpha}          \def\da{{\dot\alpha}\lambda} \def\L{\Lambda} \def\la{\lambda^\alpha}          \def\da{{\dot\alpha}$ may then describe a quantized compact $2n$-dimensional symplectic space $\cK$. 
Hence, the background can be interpreted as higher-dimensional space ${\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O} = {\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}^4 \times \cK$. 
It is well-known that such $\cK$ may indeed arise from non-Abelian fields on ${\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}^4$ via the Higgs effect, e.g. the fuzzy sphere $S^2_N$~\cite{Madore:1991,Myers:1999ps,Alekseev:1999,Aschieri:2006uw}. 
Some preliminary studies in this direction have been recently done in Ref.~\cite{Blaschke:2011qu}. 
In particular, the effective action on ${\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}^4\times\cK_N$ in the presence of fuzzy extra dimensions was studied at different scales: 
Supersymmetry can be broken by the extra dimensions $\cK_N$ and their Kaluza-Klein modes, and 
finite gravitational terms are generically induced in the trace--$U(1)$ sector. 
Maximal supersymmetry is restored above a certain scale, ensuring a UV finite effective action. 



\section{Conclusion}
We have presented a brief summary of recent results concerning the effective action of the IKKT resp. the IIB model~\cite{Ishibashi:1996xs} in the presence of dynamical SUSY breaking. 
These should illustrate (and to some extent support) the idea of a brane-world scenario with compactified extra dimensions and emergent gravity on the branes. 
Further details may be found in Refs.~\cite{Blaschke:2011qu,Chatzistavrakidis:2011gs,Steinacker:2011wb}. 
Future work on this subject will surely shed more light on this model, as there is much more to study, especially concerning compactified brane solutions of type ${\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}^4\times\cK$. 




\subsection*{Acknowledgements}
Many thanks go to the organizers of the 7$^{\rm th}$ International Conference on Quantum Theory and Symmetries in Prague, which was a wonderful and stimulating conference. 
D.N. Blaschke is a recipient of an APART fellowship of