String Branchings on Complex Tori and Algebraic Representations of Generalized Krichever-Novikov Algebras

The propagation differential for bosonic strings on a complex torus with three symmetric punctures is investigated. We study deformation aspects between two point and three point differentials as well as the behaviour of the corresponding Krichever-Novikov algebras. The structure constants are calculated and from this we derive a central extension of the Krichever-Novikov algebras by means of b-c systems. The defining cocycle for this central extension deforms to the well known Virasoro cocycle for certain kinds of degenerations of the torus. AMS subject classification (1991): 17B66, 17B90, 14H52, 30F30, 81T40Comment: 11 pages, amste

Searching for overturning convection in penumbral filaments: slit spectroscopy at 0.2 arcsec resolution

Recent numerical simulations of sunspots suggest that overturning convection is responsible for the existence of penumbral filaments and the Evershed flow, but there is little observational evidence of this process. Here we carry out a spectroscopic search for small-scale convective motions in the penumbra of a sunspot located 5 deg away from the disk center. The position of the spot is very favorable for the detection of overturning downflows at the edges of penumbral filaments. Our analysis is based on measurements of the Fe I 709.0 nm line taken with the Littrow spectrograph of the Swedish 1 m Solar Telescope under excellent seeing conditions. We compute line bisectors at different intensity levels and derive Doppler velocities from them. The velocities are calibrated using a nearby telluric line, with systematic errors smaller than 150 m/s. Deep in the photosphere, as sampled by the bisectors at the 80%-88% intensity levels, we always observe blueshifts or zero velocities. The maximum blueshifts reach 1.2 km/s and tend to be cospatial with bright penumbral filaments. In the line core we detect blueshifts for the most part, with small velocities not exceeding 300 m/s. Redshifts also occur, but at the level of 100-150 m/s, and only occasionally. The fact that they are visible in high layers casts doubts on their convective origin. Overall, we do not find indications of downflows that could be associated with overturning convection at our detection limit of 150 m/s. Either no downflows exist, or we have been unable to observe them because they occur beneath tau=1 or the spatial resolution/height resolution of the measurements is still insufficient.Comment: Accepted for publication in Ap

Penumbral thermal structure below the visible surface

$Context$. The thermal structure of the penumbra below its visible surface (i.e., $\tau_5 \ge 1$) has important implications for our present understanding of sunspots and their penumbrae: their brightness and energy transport, mode conversion of magneto-acoustic waves, sunspot seismology, and so forth. $Aims$. We aim at determining the thermal stratification in the layers immediately beneath the visible surface of the penumbra: $\tau_5 \in [1,3]$ ($\approx 70-80$ km below the visible continuum-forming layer). $Methods$. We analyzed spectropolarimetric data (i.e., Stokes profiles) in three Fe \textsc{i} lines located at 1565 nm observed with the GRIS instrument attached to the 1.5-meter solar telescope GREGOR. The data are corrected for the smearing effects of wide-angle scattered light and then subjected to an inversion code for the radiative transfer equation in order to retrieve, among others, the temperature as a function of optical depth $T(\tau_5)$. $Results$. We find that the temperature gradient below the visible surface of the penumbra is smaller than in the quiet Sun. This implies that in the region $\tau_5 \ge 1$ the penumbral temperature diverges from that of the quiet Sun. The same result is obtained when focusing only on the thermal structure below the surface of bright penumbral filaments. We interpret these results as evidence of a thick penumbra, whereby the magnetopause is not located near its visible surface. In addition, we find that the temperature gradient in bright penumbral filaments is lower than in granules. This can be explained in terms of the limited expansion of a hot upflow inside a penumbral filament relative to a granular upflow, as magnetic pressure and tension forces from the surrounding penumbral magnetic field hinder an expansion like this.Comment: 5 pages; 2 figures; accepted for publication in Astronomy and Astrophysics Letter

Absolute velocity measurements in sunspot umbrae

In sunspot umbrae, convection is largely suppressed by the strong magnetic field. Previous measurements reported on negligible convective flows in umbral cores. Based on this, numerous studies have taken the umbra as zero reference to calculate Doppler velocities of the ambient active region. To clarify the amount of convective motion in the darkest part of umbrae, we directly measured Doppler velocities with an unprecedented accuracy and precision. We performed spectroscopic observations of sunspot umbrae with the Laser Absolute Reference Spectrograph (LARS) at the German Vacuum Tower Telescope. A laser frequency comb enabled the calibration of the high-resolution spectrograph and absolute wavelength positions. A thorough spectral calibration, including the measurement of the reference wavelength, yielded Doppler shifts of the spectral line Ti i 5713.9 {\AA} with an uncertainty of around 5 m s-1. The measured Doppler shifts are a composition of umbral convection and magneto-acoustic waves. For the analysis of convective shifts, we temporally average each sequence to reduce the superimposed wave signal. Compared to convective blueshifts of up to -350 m s-1 in the quiet Sun, sunspot umbrae yield a strongly reduced convective blueshifts around -30 m s-1. {W}e find that the velocity in a sunspot umbra correlates significantly with the magnetic field strength, but also with the umbral temperature defining the depth of the titanium line. The vertical upward motion decreases with increasing field strength. Extrapolating the linear approximation to zero magnetic field reproduces the measured quiet Sun blueshift. Simply taking the sunspot umbra as a zero velocity reference for the calculation of photospheric Dopplergrams can imply a systematic velocity error.Comment: 10 pages, 7 figures, 2 tables, Appendix with 5 figure

Evolution of a magnetic flux tube in a sunspot penumbra

The motion of an individual magnetic flux tube inside the penumbra of a sunspot is studied numerically. Here, we present preliminary results. The thin flux tube approximation together with a simplified radiative heat exchange with the surroundings is used to study the evolution of a flux tube embedded into a background given by a global magneto-static sunspot model. The investigation is undertaken in order to verify the conjecture that convection in sunspot penumbrae occurs by an interchange of magnetic flux tubes. The code being developed can be used to study dynamic aspects of filamentary structure in the penumbra: the temporal and spatial fluctuations of the temperature and the magnetic field, the motion of bright penumbral grains, or the Evershed effect. Here we present the evolution of a wave formed by the tube whose fragment emerges in the penumbral photosphere and migrates towards the umbra. The properties of this wave show qualitative features of the observed bright penumbral grains with corresponding upward velocity and its correlation with brightness and the inclination of the magnetic field, and also of the Evershed effect