Higgs bundles and higher Teichm\"uller spaces

This paper is a survey on the role of Higgs bundle theory in the study of higher Teichm\"uller spaces. Recall that the Teichm\"uller space of a compact surface can be identified with a certain connected component of the moduli space of representations of the fundamental group of the surface into $\mathrm{PSL}(2,{\mathbb{R}})$. Higher Teichm\"uller spaces correspond to special components of the moduli space of representations when one replaces $\mathrm{PSL}(2,{\mathbb{R}})$ by a real non-compact semisimple Lie group of higher rank. Examples of these spaces are provided by the Hitchin components for split real groups, and maximal Toledo invariant components for groups of Hermitian type. More recently, the existence of such components has been proved for $\mathrm{SO}(p,q)$, in agreement with the conjecture of Guichard and Wienhard relating the existence of higher Teichm\"uller spaces to a certain notion of positivity on a Lie group that they have introduced. We review these three different situations, and end up explaining briefly the conjectural general picture from the point of view of Higgs bundle theory.Comment: arXiv admin note: substantial text overlap with arXiv:1511.0775

Measurement of the expansion rate of the Universe from \gamma-ray attenuation

A measurement of the expansion rate of the Universe (that is the Hubble constant, H0) is derived here using the gamma-ray attenuation observed in the spectra of gamma-ray sources produced by the interaction of extragalactic gamma-ray photons with the photons of the extragalactic background light (EBL). The Hubble constant that is determined with our technique, for a Lambda CDM cosmology, is H0=71.8_{-5.6}^{+4.6}(stat)_{-13.8}^{+7.2}(syst) km s^{-1} Mpc^{-1}. This value is compatible with present-day measurements using well established methods such as local distance ladders and cosmological probes. The recent detection of the cosmic gamma-ray horizon (CGRH) from multiwavelength observation of blazars, together with the advances in the knowledge of the EBL, allow us to measure the expansion rate of the Universe. This estimate of the Hubble constant shows that gamma-ray astronomy has reached a mature enough state to provide cosmological measurements, which may become more competitive in the future with the construction of the Cherenkov Telescope Array. We find that the maximum dependence of the CGRH on the Hubble constant is approximately between redshifts 0.04 and 0.1, thus this is a smoking gun for planning future observational efforts. Other cosmological parameters, such as the total dark matter density Omega_m and the dark energy equation of state w, are explored as well.Comment: 6 pages, 3 figures; accepted by ApJ Letters on June 10, 201

A Polygonal Discontinuous Galerkin Method with Minus One Stabilization

We propose an Hybridized Discontinuous Galerkin method on polygonal tessellations, stabilized by penalizing, locally in each element $K$, a residual term involving the fluxes, measured in the norm of the dual of $H^1(K)$. The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space of VEM type. Stability and optimal error estimates in the broken $H^1$ norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.Comment: 27 pages, 2 figure

New Parametrizations of Non-Gaussian Line-of-sight Velocity Distribution

A five-parameter fitting formula for the line-of-sight stellar velocity distributions of steady state systems is proposed. It can faithfully reproduce velocity distributions of theoretical models ranging from nearly Gaussian profiles to strongly skewed or mildly double-peaked profiles. In contrast to van der Marel and Franx (1993) and Kuijken and Merrifield (1993), the line profiles are required to have neither multi-peaks nor negative velocity wings. Information of the profile is mostly specified by five physically meaningful and nearly orthogonal fitting parameters.Comment: submitted to MNRAS; 22 pages with 3 tables and 8 figures in uuencoded compressed PS file. Also available at ftp://ibm-1.mpa-garching.mpg.de/pub/hsz/profile.u

Higgs bundles for the non-compact dual of the unitary group

Using Morse-theoretic techniques, we show that the moduli space of U*(2n)-Higgs bundles over a compact Riemann surface is connected.Comment: 20 pages; v2: several improvements and corrections; main results are unchange