Heat kernel expansion and induced action for matrix models

Abstract

In this proceeding note 1 , I review some recent results concerning the quantum effective action of certain matrix models, i.e. the supersymmetric IKKT model, in the context of emergent gravity. The absence of pathological UV/IR mixing is discussed, as well as dynamical SUSY breaking and some relations with string theory and supergravity. BackgroundOne of the greatest challenges of present day physics is to find a mathematical framework to describe quantum field theory formulated on some quantum space-time. This would be necessary in order to combine the standard model of particle physics with gravity. Surely, many ideas are currently around and it remains unclear which is the best way to achieve this goal. What I would like to present in this short proceeding note, are some recent results [1,2] which are based on the idea of gravity as an emergent force derived from matrix models -for a general introduction to the subject, the interested reader is referred to [3].It is indeed not easy to find a new model that is renormalizable, but a concept that is known to greatly improve the properties of a model with respect to divergences and renormalizability is supersymmetry. Incidentally, a supersymmetric matrix model which turns out to achieve what we want, namely incorporating scalars, fermions, gauge bosons and (emergent) gravity was originally introduced as a non-perturbative description of type IIB string theory: It is the so-called IKKT model, named after its authors [4], and given by the actionwhere X a , a = 0, 1, 2, . . . , 9 are Hermitian matrices, Ψ is a matrix-valued Majorana-Weyl spinor of SO(9, 1), and the γ a form the corresponding Clifford algebra. The model is obtained by dimensional reduction of the 10-dimensional SU (N ) Super-Yang-Mills theory to a point, and * daniel.blaschke@univie.ac.at 1 based on a talk given at the 7 th International Conference on Quantum Theory and Symmetries, August 7-13, 2011 in Prague, Czech Republic 1 arXiv:1111.0876v1 [hep-th] 3 Nov 2011 taking the N → ∞ limit. Indices are raised and lowered using the fixed background metric g ab , which we consider to either be the ten-dimensional Minkowski or Euclidean metric. The action (1) is invariant under the N = 2 matrix supersymmetryas well as the additional symmetrieswhere the tilde indicates the corresponding spin group. Although an explicit proof to all orders is still missing, there are good reasons to believe that this model is in fact renormalizable [4,5].Here we take the IKKT matrix model model (1) as a starting point independent of string theory, and focus on 4-dimensional brane solutions or backgrounds, considered as physical space(-time). Due to supersymmetry, the model should provide a well-defined quantum theory at least for 4-dimensional backgrounds. In fact, numerical evidence for the emergence of 3+1-dimensional space-time within the IKKT model has been obtained recently in Ref. [6], providing further motivation to study the effective physics of 4-dimensional backgrounds.Additionally, it was shown in [7] ho...