New Algorithm and Phase Diagram of Noncommutative Phi**4 on the Fuzzy Sphere

We propose a new algorithm for simulating noncommutative phi-four theory on the fuzzy sphere based on, i) coupling the scalar field to a U(1) gauge field, in such a way that in the commutative limit N\longrightarrow \infty, the two modes decouple and we are left with pure scalar phi-four on the sphere, and ii) diagonalizing the scalar field by means of a U(N) unitary matrix, and then integrating out the unitary group from the partition function. The number of degrees of freedom in the scalar sector reduces, therefore, from N^2 to the N eigenvalues of the scalar field, whereas the dynamics of the U(1) gauge field, is given by D=3 Yang-Mills matrix model with a Myers term. As an application, the phase diagram, including the triple point, of noncommutative phi-four theory on the fuzzy sphere, is reconstructed with small values of N up to N=10, and large numbers of statistics.Comment: 29 pages,9 figures, 4 tables, v2: new section added in which we compare briefly between the different algorithms,30 pages, v3:two figures added, one equation added, various comments added throughout the article, typos corrected, writing style improved, 33 page

Quantum Equivalence of NC and YM Gauge Theories in 2 D and Matrix Theory

We construct noncommutative U(1) gauge theory on the fuzzy sphere S^2_N as a unitary 2N x 2N matrix model. In the quantum theory the model is equivalent to a nonabelian U(N) Yang-Mills theory on a 2 dimensional lattice with 2 plaquettes. This equivalence holds in the " fuzzy sphere" phase where we observe a 3rd order phase transition between weak-coupling and strong-coupling phases of the gauge theory. In the ``matrix'' phase we have a U(N) gauge theory on a single point.Comment: 13 pages, one grap

Fuzzy Non-Trivial Gauge Configurations

In this talk we will report on few results of discrete physics on the fuzzy sphere . In particular non-trivial field configurations such as monopoles and solitons are constructed on fuzzy ${\bf S}^2$ using the language of K-theory, i.e projectors . As we will show, these configurations are intrinsically finite dimensional matrix models . The corresponding monopole charges and soliton winding numbers are also found using the formalism of noncommutative geometry and cyclic cohomology .Comment: 9 pages . Talk delivered in the MRST 2001 conference, University of Western Ontario, London, Ontario . To be published in the conference proceeding

Matrix Model Fixed Point of Noncommutative Phi-Four

In this article we exhibit explicitly the matrix model ($\theta=\infty$) fixed point of phi-four theory on noncommutative spacetime with only two noncommuting directions using the Wilson renormalization group recursion formula and the 1/N expansion of the zero dimensional reduction and then calculate the mass critical exponent $\nu$ and the anomalous dimension $\eta$ in various dimensions .Comment: v3: 51 pages, section 3 is enlarged further, section 6 on the Grosse-Vignes-Tourneret model is new, a new appendix is added, the rest is unchange

On the Problem of Vacuum Energy in FLRW Universes and Dark Energy

We present a (hopefully) novel calculation of the vacuum energy in expanding FLRW spacetimes based on the renormalization of quantum field theory in non-zero backgrounds. We compute the renormalized effective action up to the $2-$point function and then apply the formalism to the cosmological backgrounds of interest. As an example we calculate for quasi de Sitter spacetimes the leading correction to the vacuum energy given by the tadpole diagram and show that it behaves as $\sim H_0^2 \Lambda_{\rm pl}$ where $H_0$ is the Hubble constant and $\Lambda_{\rm pl}$ is the Planck constant. This is of the same order of magnitude as the observed dark energy density in the universe.Comment: 5 pages, 1 figure; v2: reorganization of the presentation and minor changes and comments adde

Topology Change From Quantum Instability of Gauge Theory on Fuzzy CP^2

Many gauge theory models on fuzzy complex projective spaces will contain a strong instability in the quantum field theory leading to topology change. This can be thought of as due to the interaction between spacetime via its noncommutativity and the fields (matrices) and it is related to the perturbative UV-IR mixing. We work out in detail the example of fuzzy CP^2 and discuss at the level of the phase diagram the quantum transitions between the 3 spaces (spacetimes) CP^2, S^2 and the 0-dimensional space consisting of a single point {0}.Comment: 26 pages, one grap