The exactly solvable spin Sutherland model of B_N type and its related spin chain

We compute the spectrum of the su(m) spin Sutherland model of B_N type, including the exact degeneracy of all energy levels. By studying the large coupling constant limit of this model and of its scalar counterpart, we evaluate the partition function of their associated spin chain of Haldane-Shastry type in closed form. With the help of the formula for the partition function thus obtained we study the chain's spectrum, showing that it cannot be obtained as a limiting case of its BC_N counterpart. The structure of the partition function also suggests that the spectrum of the Haldane-Shastry spin chain of B_N type is equivalent to that of a suitable vertex model, as is the case for its A_{N-1} counterpart, and that the density of its eigenvalues is normally distributed when the number of sites N tends to infinity. We analyze this last conjecture numerically using again the explicit formula for the partition function, and check its validity for several values of N and m.Comment: Typeset in LaTeX (24 pages, 4 figures). arXiv admin note: text overlap with arXiv:0909.296

The effect of the linear term on the wavelet estimator of primordial non-Gaussianity

In this work we present constraints on different shapes of primordial non-Gaussianity using the Wilkinson Microwave Anisotropy Probe (WMAP) 7-year data and the spherical Mexican hat wavelet fnl estimator including the linear term correction. In particular we focus on the local, equilateral and orthogonal shapes. We first analyse the main statistical properties of the wavelet estimator and show the conditions to reach optimality. We include the linear term correction in our estimators and compare the estimates with the values already published using only the cubic term. The estimators are tested with realistic WMAP simulations with anisotropic noise and the WMAP KQ75 sky cut. The inclusion of the linear term correction shows a negligible improvement (< 1 per cent) in the error-bar for any of the shapes considered. The results of this analysis show that, in the particular case of the wavelet estimator, the optimality for WMAP anisotropy levels is basically achieved with the mean subtraction and in practical terms there is no need of including a linear term once the mean has been subtracted. Our best estimates are now: local fnl = 39.0 +/ 21.4, equilateral fnl = -62.8 +/- 154.0 and orthogonal fnl = -159.8 +/- 115.1 (all cases 68 per cent CL). We have also computed the expected linear term correction for simulated Planck maps with anisotropic noise at 143 GHz following the Planck Sky Model and including a mask. The improvement achieved in this case for the local fnl error-bar is also negligible (0.4 per cent).Comment: 8 pages, 5 figures, 4 tables. Minor revision, one figure added, accepted for publication in MNRA

Monomial transformations of the projective space

We prove that, over any field, the dimension of the indeterminacy locus of a rational transformation $f$ of $P^n$ which is defined by monomials of the same degree $d$ with no common factors is at least $(n-2)/2$, provided that the degree of $f$ as a map is not divisible by $d$. This implies upper bounds on the multidegree of $f$

Duality for massive spin two theories in arbitrary dimensions

Using the parent Lagrangian approach we construct a dual formulation, in the sense originally proposed by Curtright and Freund, of a massive spin two Fierz-Pauli theory in arbitrary dimensions $D$. This is achieved in terms of a mixed symmetry tensor $T_{A[B_{1}B_{2}... B_{D-2}]}$, without the need of auxiliary fields. The relation of this method with an alternative formulation based on a gauge symmetry principle proposed by Zinoviev is elucidated. We show that the latter formulation in four dimensions, with a given gauge fixing together with a definite sequence of auxiliary fields elimination via their equations of motion, leads to the parent Lagrangian already considered by West completed by a Fierz-Pauli mass term, which in turns yields the Curtright-Freund action. This motivates our generalization to arbitrary dimensions leading to the corresponding extension of the four dimensional result. We identify the transverse true degrees of freedom of the dual theory and verify that their number is in accordance with those of the massive Fierz-Pauli field.Comment: 14 pages, no figures, revtex 4. Some new comments and explanations have been added and the notation homogeneize

Few-anyon systems in a parabolic dot

The energy levels of two and three anyons in a two-dimensional parabolic quantum dot and a perpendicular magnetic field are computed as power series in 1/|J|, where J is the angular momentum. The particles interact repulsively through a coulombic (1/r) potential. In the two-anyon problem, the reached accuracy is better than one part in 10^5. For three anyons, we study the combined effects of anyon statistics and coulomb repulsion in the ``linear'' anyonic states.Comment: LaTeX, 6 pages, 4 postscript figure

Rational quantum integrable systems of D_N type with polarized spin reversal operators

We study the spin Calogero model of D_N type with polarized spin reversal operators, as well as its associated spin chain of Haldane-Shastry type, both in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and the partition function of the former model in closed form, from which we derive an exact formula for the chain's partition function in terms of products of partition functions of Polychronakos-Frahm spin chains of type A. Using a recursion relation for the latter partition functions that we derive in the paper, we are able to numerically evaluate the partition function, and thus the spectrum, of the D_N-type spin chain for relatively high values of the number of spins N. We analyze several global properties of the chain's spectrum, such as the asymptotic level density, the distribution of consecutive spacings of the unfolded spectrum, and the average degeneracy. In particular, our results suggest that this chain is invariant under a suitable Yangian group, and that its spectrum coincides with that of a Yangian-invariant vertex model with linear energy function and dispersion relation.Comment: 26 pages, 5 figures, typeset in LaTe