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

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