Critical Behaviour of the Fuzzy Sphere

We study a multi-matrix model whose low temperature phase is a fuzzy sphere that undergoes an evaporation transition as the temperature is increased. We investigate finite size scaling of the system as the limiting temperature of stability of the fuzzy sphere phase is approached. We find on theoretical grounds that the system should obey scaling with specific heat exponent \alpha=1/2, shift exponent \bar \lambda=4/3 and that the peak in the specific heat grows with exponent \bar \omega=2/3. Using hybrid Monte Carlo simulations we find good collapse of specific heat data consistent with a scaling ansatz which give our best estimates for the scaling exponents as \alpha=0.50 \pm 0.01,\bar \lambda=1.41 \pm 0.08 and \bar \omega=0.66 \pm 0.08 .Comment: 30 pages, 10 figure

Scalar and Spinor Field Actions on Fuzzy S4S^4: fuzzy CP3CP^3 as a SF2S^2_F bundle over SF4S^4_F

We present a manifestly Spin(5) invariant construction of squashed fuzzy $CP^3$ as a fuzzy $S^2$ bundle over fuzzy $S^4$. We develop the necessary projectors and exhibit the squashing in terms of the radii of the $S^2$ and $S^4$. Our analysis allows us give both scalar and spinor fuzzy action functionals whose low lying modes are truncated versions of those of a commutative $S^4$.Comment: 19 page

A Fuzzy Three Sphere and Fuzzy Tori

A fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces, of complex dimension one and three, by modifying the Laplacians on the latter so as to give unwanted states large eigenvalues. This leaves only states corresponding to fuzzy spheres in the low energy spectrum. The construction of a fuzzy circle opens the way to fuzzy tori of any dimension, thus circumventing the problem of power law corrections in possible numerical simulations on these spaces

Countdown to 2010: Can we assess Ireland’s insect species diversity and loss?

peer-reviewedThe insects are the most diverse organisms on this planet and play an essential role in ecosystem functioning, yet we know very little about them. In light of the Convention on Biological Diversity, this paper summarises the known insect species numbers for Ireland and questions whether this is a true refl ection of our insect diversity. The total number of known species for Ireland is 11,422. Using species accumulation curves and a comparison with the British fauna, this study shows that the Irish list is incomplete and that the actual species number is much higher. However, even with a reasonable knowledge of the species in Ireland, insects are such speciose, small, and inconspicuous animals that it is diffi cult to assess species loss. It is impossible to know at one point in time the number of insect species in Ireland and, although it is useful to summarise the known number of species, it is essential that biodiversity indicators, such as the Red List Index, are developed

A Fuzzy Three Sphere and Fuzzy Tori

A fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces, of complex dimension one and three, by modifying the Laplacians on the latter so as to give unwanted states large eigenvalues. This leaves only states corresponding to fuzzy spheres in the low energy spectrum (this allows the commutative algebra of functions on the continuous sphere to be approximated to any required degree of accuracy). The construction of a fuzzy circle opens the way to fuzzy tori of any dimension, thus circumventing the problem of power law corrections in possible numerical simulations on these spaces

Matrix φ^4 models on the fuzzy sphere and their continuum limits

We demonstrate that the UV/IR mixing problems found recently for a scalar φ^4 theory on the fuzzy sphere are localized to tadpole diagrams and can be overcome by a suitable modification of the action. This modification is equivalent to normal ordering the φ^4 vertex. In the limit of the commutative sphere, the perturbation theory of this modified action matches that of the commutative theory

Fuzzy Complex Quadrics and Spheres

A matrix algebra is constructed which consists of the necessary degrees of freedom for a finite approximation to the algebra of functions on the family of orthogonal Grassmannians of real dimension 2N, known as complex quadrics. These matrix algebras contain the relevant degrees of freedom for describing truncations of harmonic expansions of functions on N-spheres. An Inonu-Wigner contraction of the quadric gives the co-tangent bundle to the commutative sphere in the continuum limit. It is shown how the degrees of freedom for the sphere can be projected out of a finite dimensional functional integral, using second-order Casimirs, giving a well-defined procedure for construction functional integrals over fuzzy spheres of any dimension

Scalar and spinor field actions on fuzzy S4: fuzzy CP3 as a S2F bundle over S4F

We present a manifestly Spin(5) invariant construction of s quashed fuzzy CP3 as a fuzzy S2 bundle over fuzzy S4 . We develop the necessary projectors and exhibit the squashing in terms of the radii of the S2 and S4 . Our analysis allows us give both scalar and spinor fuzzy action functionals whose low lying modes are truncated versions of those of a commutative S4