TOPICAL REVIEW: Emergent geometry and gravity from matrix models: an introduction

An introductory review to emergent noncommutative gravity within Yang-Mills matrix models is presented. Spacetime is described as a noncommutative brane solution of the matrix model, i.e. as a submanifold of {\mathbb 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 non-commutative gauge theory and the role of UV/IR mixing are explained. Several types of geometries are identified, in particular 'harmonic' and 'Einstein' types of solutions. The physics of the harmonic branch is discussed in some detail, emphasizing the non-standard role of vacuum energy. This may provide a 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 quantum theory of fundamental interactions including gravity.