Towards an exact frame formulation of conformal higher spins in three dimensions

In this note we discuss some aspects of the frame formulation of conformal higher spins in three dimensions. We give some exact formulae for the coupled spin two - spin three part of the full higher spin theory and propose a star product Lagrangian for all spins from two and up. Since there is no consistent Lagrangian formulation based on the Poisson bracket we start the construction from the field equations in this approximation of the star product. The higher spin algebra is then realized in terms of classical variables which leads to certain important simplifications that we take advantage of. The suggested structure of the all-spin Lagrangian given here is, however, obtained using an expansion of the star product beyond the Poisson bracket in terms of multi-commutators and the Lagrangian should be viewed as a starting point for the derivation of the full theory based on a star product. How to do this is explained as well as how to include the coupling to scalar fields. We also comment on the AdS/CFT relation to four dimensions.Comment: 18 pages, v2: misprints corrected, an appendix, footnotes and some clarifying remarks added, 21 page

A New Noncommutative Product on the Fuzzy Two-Sphere Corresponding to the Unitary Representation of SU(2) and the Seiberg-Witten Map

We obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a unitary representation. This is done by constructing a star product, $\star_{\lambda}$, for an arbitrary representation of SU(2) which depends on a continuous parameter $\lambda$ and searching for the values of $\lambda$ which give unitary representations. We will find two series of values: $\lambda = \lambda^{(A)}_j=1/(2j)$ and $\lambda=\lambda^{(B)}_j =-1/(2j+2)$, where j is the spin of the representation of SU(2). At $\lambda = \lambda^{(A)}_j$ the new star product $\star_{\lambda}$ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order $\ell \leq 2j$ and then $\star_{\lambda}$ reduces to the star product $\star$ obtained by Preusnajder. The star product at $\lambda=\lambda^{(B)}_j$, to be denoted by $\bullet$, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order $\ell \leq 2j$. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg-Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg-Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point $\lambda=0$.Comment: 15 pages, typos corrected, references added, a note adde

Quantum field theory in generalised Snyder spaces

We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate peturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1608.0620

Convolution of orbital measures on symmetric spaces of type CpC_p and DpD_p

We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star\delta_{e^Y}^\natural$ of two orbital measures on the symmetric spaces ${\bf SO}_0(p,p)/{\bf SO}(p)\times{\bf SO}(p)$, \SU(p,p)/{\bf S}({\bf U}(p)\times{\bf U}(p)) and \Sp(p,p)/{\bf Sp }(p)\times\Sp(p). We prove sharp conditions on $X$, Y\in\a for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.Comment: arXiv admin note: text overlap with arXiv:1212.000

Fast Recognition of Partial Star Products and Quasi Cartesian Products

This paper is concerned with the fast computation of a relation $\R$ on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of $\R$ is the relation $\delta^\ast$, whose convex closure yields the product relation $\sigma$ that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of $\R$ so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of $\delta^\ast$ for graphs with maximum bounded degree. Furthermore, we define \emph{quasi Cartesian products} as graphs with non-trivial $\delta^\ast$. We provide several examples, and show that quasi Cartesian products can be recognized in linear time for graphs with bounded maximum degree. Finally, we note that quasi products can be recognized in sublinear time with a parallelized algorithm