The tetrahexahedric angular Calogero model

The spherical reduction of the rational Calogero model (of type $A_{n-1}$ and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the $(n{-}2)$-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, $n{=}4$, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.Comment: 1+29 pages, 4 figures; v2: Introduction extended, eq.(5.16) and one ref. added, published versio

Thermodynamics of Lovelock black holes with a nonminimal scalar field

We source the Lovelock gravity theories indexed by an integer k and fixed by requiring a unique anti-de Sitter vacuum with a self-interacting nonminimal scalar field in arbitrary dimension d. For each inequivalent Lovelock gravity theory indexed by the integer k, we establish the existence of a two-parametric self-interacting potential that permits to derive a class of black hole solutions with planar horizon for any arbitrary value of the nonminimal coupling parameter. In the thermodynamical analysis of the solution, we show that, once regularized the Euclidean action, the mass contribution coming form the gravity side exactly cancels, order by order, the one arising from the matter part yielding to a vanishing mass. This result is in accordance with the fact that the entropy of the solution, being proportional to the lapse function evaluated at the horizon, also vanishes. Consequently, the integration constant appearing in the solution is interpreted as a sort of hair which turns out to vanish at high temperature.Comment: 11 page

PT{\cal PT} deformation of angular Calogero models

The rational Calogero model based on an arbitrary rank-$n$ Coxeter root system is spherically reduced to a superintegrable angular model of a particle moving on $S^{n-1}$ subject to a very particular potential singular at the reflection hyperplanes. It is outlined how to find conserved charges and to construct intertwining operators. We deform these models in a ${\cal PT}$-symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The ${\cal PT}$ deformation does not change the energy eigenvalues but in some cases adds a previously unphysical tower of states. For integral couplings the new and old energy levels coincide, which roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry charge. We present the details for the generic rank-two ($A_2$, $G_2$) and all rank-three Coxeter systems ($AD_3$, $BC_3$ and $H_3$), including a reducible case ($A_1^{\otimes 3}$).Comment: 1+41 pages, 12 figure

Nonlinear supersymmetry in the quantum Calogero model

It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g+1. For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order n(n-1)(2g-1)/2 in the momenta and squares to a polynomial in the Liouville charges. With a natural Z_2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.Comment: 1+21 pages; v2: minor corrections, typos in eq.(5.18) fixed, 3 refs. added, published versio