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

String states, loops and effective actions in noncommutative field theory and matrix models

Refining previous work by Iso, Kawai and Kitazawa, we discuss bi-local string states as a tool for loop computations in noncommutative field theory and matrix models. Defined in terms of coherent states, they exhibit the stringy features of noncommutative field theory. This leads to a closed form for the 1-loop effective action in position space, capturing the long-range non-local UV/IR mixing for scalar fields. The formalism applies to generic fuzzy spaces. The non-locality is tamed in the maximally supersymmetric IKKT or IIB model, where it gives rise to supergravity. The linearized supergravity interactions are obtained directly in position space at one loop using string states on generic noncommutative branes.Comment: 31 pages, 2 figure