Shintani functions, real spherical manifolds, and symmetry breaking operators

For a pair of reductive groups $G \supset G'$, we prove a geometric criterion for the space $Sh(\lambda, \nu)$ of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs $(G,G')$ having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of $dim Sh(\lambda, \nu)$ is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for $(G, G') = (O(n+1,1), O(n,1))$.Comment: to appear in Progress in Mathematics, Birkhause

X-ray background and its correlation with the 21 cm signal

We use high resolution hydrodynamical simulations to study the contribution to the X-ray background from high-$z$ energetic sources, such as X-ray binaries, accreting nuclear black holes and shock heated interstellar medium. Adopting the model discussed in Eide et al. (2018), we find that these X-ray sources during the Epoch of Reionization (EoR) contribute less than a few percent of the unresolved X-ray background. The same sources contribute to less than $\sim$2\% of the measured angular power spectrum of the fluctuations of the X-ray background. The outputs of radiative transfer simulations modeling the EoR are used to evaluate the cross-correlations of X-ray background with the 21~cm signal from neutral hydrogen. Such correlation could be used to confirm the origin of the 21 cm signal, as well as give information on the properties of the X-ray sources during the EoR. We find that the correlations are positive during the early stages of reionization when most of the hydrogen is neutral, while they become negative when the intergalactic medium gets highly ionized, with the transition from positive to negative depending on both the X-ray model and the scale under consideration. With {\tt SKA} as the reference instrument for the 21~cm experiment, the predicted S/N for such correlations is $<1$ if the corresponding X-ray survey is only able to resolve and remove X-ray sources with observed flux $>10^{-15}\,\rm erg\, cm^{-2} \, s^{-1}$, while the cumulative S/N from $l=1000$ to $10^{4}$ at $x_{\rm HI}=0.5$ is $\sim 5$ if sources with observed flux $>10^{-17}\,\rm erg\, cm^{-2} \, s^{-1}$ are detected.Comment: 9 pages, 8 figure

Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

Wigner transform and pseudodifferential operators on symmetric spaces of non-compact type

We obtain a general expression for a Wigner transform (Wigner function) on symmetric spaces of non-compact type and study the Weyl calculus of pseudodifferential operators on them

On Kostant's partial order on hyperbolic elements

We study Kostant's partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements.Comment: 7 page

Measuring patchy reionisation with kSZ2^2-21 cm correlations

We study cross-correlations of the kinetic Sunyaev-Zel'dovich effect (kSZ) and 21 cm signals during the epoch of reionisation (EoR) to measure the effects of patchy reionisation. Since the kSZ effect is proportional to the line-of-sight velocity, the kSZ-21 cm cross correlation suffers from cancellation at small angular scales. We thus focus on the correlation between the kSZ-squared field (kSZ$^2$) and 21 cm signals. When the global ionisation fraction is low ($x_e\lesssim 0.7$), the kSZ$^2$ fluctuation is dominated by rare ionised bubbles which leads to an anti-correlation with the 21 cm signal. When $0.8\lesssim x_e<1$, the correlation is dominated by small pockets of neutral regions, leading to a positive correlation. However, at very high redshifts when $x_e<0.15$, the spin temperature fluctuations change the sign of the correlation from negative to positive, as weakly ionised regions can have strong 21 cm signals in this case. To extract this correlation, we find that Wiener filtering is effective in removing large signals from the primary CMB anisotropy. The expected signal-to-noise ratios for a $\sim$10-hour integration of upcoming Square Kilometer Array data cross-correlated with maps from the current generation of CMB observatories with 3.4~$\mu$K arcmin noise and 1.7~arcmin beam over 100~deg$^2$ are 51, 60, and 37 for $x_e=0.2$, 0.5, and 0.9, respectively.Comment: 7pages, 7 figure

Magic Supergravities, N= 8 and Black Hole Composites

We present explicit U-duality invariants for the R, C, Q, O$ (real, complex, quaternionic and octonionic) magic supergravities in four and five dimensions using complex forms with a reality condition. From these invariants we derive an explicit entropy function and corresponding stabilization equations which we use to exhibit stationary multi-center 1/2 BPS solutions of these N=2 d=4 theories, starting with the octonionic one with E_{7(-25)} duality symmetry. We generalize to stationary 1/8 BPS multicenter solutions of N=8, d=4 supergravity, using the consistent truncation to the quaternionic magic N=2 supergravity. We present a general solution of non-BPS attractor equations of the STU truncation of magic models. We finish with a discussion of the BPS-non-BPS relations and attractors in N=2 versus N= 5, 6, 8.Comment: 33 pages, references added plus brief outline at end of introductio

Extremal Black Attractors in 8D Maximal Supergravity

Motivated by the new higher D-supergravity solutions on intersecting attractors obtained by Ferrara et al. in [Phys.Rev.D79:065031-2009], we focus in this paper on 8D maximal supergravity with moduli space [SL(3,R)/SO(3)]x[SL(2,R)/SO(2)] and study explicitly the attractor mechanism for various configurations of extremal black p- branes (anti-branes) with the typical near horizon geometries AdS_{p+2}xS^{m}xT^{6-p-m} and p=0,1,2,3,4; 2<=m<=6. Interpretations in terms of wrapped M2 and M5 branes of the 11D M-theory on 3-torus are also given. Keywords: 8D supergravity, black p-branes, attractor mechanism, M-theory.Comment: 37 page

Symmetric spaces of higher rank do not admit differentiable compactifications

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.Comment: 13 pages, to appear in Mathematische Annale

Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with computer algebra systems. As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified. The submission also contains an example implementation of the algorithms and formulas of the paper as a package for Maple 10, the technical documentation for this implementation, and a worksheet carrying out the computations for the space SU(3)/SO(3) used in the proof of Proposition 6.1 of the paper.Comment: 23 pages, also contains two Maple worksheets and technical documentatio