Split noncommutativity and compactified brane solutions in matrix models

Solutions of the undeformed IKKT matrix model with structure R^{3,1} x K are presented, where the noncommutativity relates the compact with the non-compact space. The extra dimensions are stabilized by angular momentum, and the scales of K are generic moduli of the solutions. Explicit solutions are given for K= T^2, K= T^4, K = S^2 x T^2 and K = S^2 x S^2. Infinite towers of Kaluza-Klein modes may arise in some directions, along with an effective UV cutoff on the non-compact space. Deformations of these solutions carry NC gauge theory coupled to (emergent) gravity. Analogous solutions of the BFSS model are also given.Comment: 24 pages. V2, V3: typos fixed. V4: minor correction

Quantization and eigenvalue distribution of noncommutative scalar field theory

The quantization of noncommutative scalar field theory is studied from the matrix model point of view, exhibiting the significance of the eigenvalue distribution. This provides a new framework to study renormalization, and predicts a phase transition in the noncommutative \phi^4 model. In 4-dimensions, the corresponding critical line is found to terminate at a non-trivial point.Comment: Talk presented at the II. Southeastern European Workshop BW 2005, Vrnjacka Banja, Serbia and the XIV. Workshop of Geometric Methods in Physics, Bialowieza, Poland. To appear in Facta Universitati

Finite dimensional unitary representations of quantum Anti-de Sitter groups at roots of unity

We study irreducible unitary \reps of $U_q(SO(2,1))$ and $U_q(SO(2,3))$ for $q$ a root of unity, which are finite dimensional. Among others, unitary \reps corresponding to all classical one-particle representations with integral weights are found for $q = e^{i \pi /M}$, with $M$ being large enough. In the "massless" case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of "pure gauges", as classically. A truncated associative tensor product describing unitary many-particle representations is defined for $q = e^{i\pi /M}$.Comment: More systematic proof of statements on the structure of irreps, some typos corrected. 25 pages LaTeX, 4 figures included using epsf. To appear in Comm. Math. Phy

A non-perturbative approach to non-commutative scalar field theory

Non-commutative Euclidean scalar field theory is shown to have an eigenvalue sector which is dominated by a well-defined eigenvalue density, and can be described by a matrix model. This is established using regularizations of R^{2n}_\theta via fuzzy spaces for the free and weakly coupled case, and extends naturally to the non-perturbative domain. It allows to study the renormalization of the effective potential using matrix model techniques, and is closely related to UV/IR mixing. In particular we find a phase transition for the \phi^4 model at strong coupling, to a phase which is identified with the striped or matrix phase. The method is expected to be applicable in 4 dimensions, where a critical line is found which terminates at a non-trivial point, with nonzero critical coupling. This provides evidence for a non-trivial fixed-point for the 4-dimensional NC \phi^4 model.Comment: 38 pages, 1 figure. V2: references added. V3: typos fixed, discussion added; published versio

One-loop stabilization of the fuzzy four-sphere via softly broken SUSY

We describe a stabilization mechanism for fuzzy $S^4_N$ in the Euclidean IIB matrix model due to vacuum energy in the presence of a positive mass term. The one-loop effective potential for the radius contains an attractive contribution attributed to supergravity, while the mass term induces a repulsive contribution for small radius due to SUSY breaking. This leads to a stabilization of the radius. The mechanism should be pertinent to recent results on the genesis of 3+1-dimensional space-time in the Minkowskian IIB model.Comment: 28 pages, 3 figures. V2: typos fixed, improved discussion, published version. V3,V4: reference adde

Cosmological space-times with resolved Big Bang in Yang-Mills matrix models

We present simple solutions of IKKT-type matrix models that can be viewed as quantized homogeneous and isotropic cosmological space-times, with finite density of microstates and a regular Big Bang (BB). The BB arises from a signature change of the effective metric on a fuzzy brane embedded in Lorentzian target space, in the presence of a quantized 4-volume form. The Hubble parameter is singular at the BB, and becomes small at late times. There is no singularity from the target space point of view, and the brane is Euclidean "before" the BB. Both recollapsing and expanding universe solutions are obtained, depending on the mass parameters.Comment: 22 pages, 4 figures. V2,V3: improved discussion, typos fixed. V3: published versio