A introductory review to emergent noncommutative gravity within Yang-Mills Matrix models is presented. Space-time is described as a noncommutative brane solution of the matrix model, i.e. as submanifold of R D . Fields and matter on the brane arise as fluctuations of the bosonic resp. fermionic matrices around such a background, and couple to an effective metric interpreted in terms of gravity. Suitable tools are provided for the description of the effective geometry in the semi-classical limit. The relation to noncommutative gauge theory and the role of UV/IR mixing is explained. Several types of geometries are identified, in particular "harmonic" and "Einstein" type of solutions. The physics of the harmonic branch is discussed in some detail, emphasizing the non-standard role of vacuum energy. This may provide new approach to some of the big puzzles in this context. The IKKT model with D = 10 and close relatives are singled out as promising candidates for a quantum theory of fundamental interactions including gravity.

U(n) Yang-Mills theory on the fuzzy sphere S 2 N is quantized using random matrix methods. The gauge theory is formulated as a matrix model for a single Hermitian matrix subject to a constraint, and a potential with two degenerate minima. This allows to reduce the path integral over the gauge fields to an integral over eigenvalues, which can be evaluated for large N. The partition function of U(n) Yang-Mills theory on the classical sphere is recovered in the large N limit, as a sum over instanton contributions. The monopole solutions are found explicitly.

We study the one loop dynamics of QFT on the fuzzy sphere and calculate the planar and nonplanar contributions to the two point function at one loop. We show that there is no UV/IR mixing on the fuzzy sphere. The fuzzy sphere is characterized by two moduli: a dimensionless parameter N and a dimensionful radius R. Different geometrical phases can obtained at different corners of the moduli space. In the limit of the commutative sphere, we find that the two point function is regular without UV/IR mixing; however quantization does not commute with the commutative limit, and a finite "noncommutative anomaly" survives in the commutative limit. In a different limit, the noncommutative plane R 2 θ is obtained, and the UV/IR mixing reappears. This provides an explanation of the UV/IR mixing as an infinite variant of the "noncommutative anomaly".

We present a renormalizable 4-dimensional SU(N ) gauge theory with a suitable multiplet of scalar fields, which dynamically develops extra dimensions in the form of a fuzzy sphere S 2 N . We explicitly find the tower of massive KaluzaKlein modes consistent with an interpretation as gauge theory on M 4 × S 2 , the scalars being interpreted as gauge fields on S 2 . The gauge group is broken dynamically, and the low-energy content of the model is determined. Depending on the parameters of the model the low-energy gauge group can be SU(n), or broken further to SU(n 1 ) × SU(n 2 ) × U(1), with mass scale determined by the size of the extra dimension.

We show that the matrix-model action for noncommutative U(n) gauge theory actually describes SU(n) gauge theory coupled to gravity. This is elaborated in the 4-dimensional case. The SU(n) gauge fields as well as additional scalar fields couple to an effective metric G ab , which is determined by a dynamical Poisson structure. The emergent gravity is intimately related to noncommutativity, encoding those degrees of freedom which are usually interpreted as U(1) gauge fields. This leads to a class of metrics which contains the physical degrees of freedom of gravitational waves, and allows to recover e.g. the Newtonian limit with arbitrary mass distribution. It also suggests a consistent picture of UV/IR mixing in terms of an induced gravity action. This should provide a suitable framework for quantizing gravity.