Very Long Baseline polarimetry and the gamma-ray connection in Markarian 421 during the broadband campaign in 2011

Context. This is the third paper in a series devoted to the analysis of the multiwavelength data from a campaign on the nearby (z = 0.03) TeV blazar Mrk 421 during 2011. Aims: We investigate the structure of the high angular resolution polarization, the magnetic topology, the total intensity light curve, the \u3b3-ray flux, and the photon index. We describe how they evolve and how they are connected. Methods: We analyzed data in polarized intensity obtained with the Very Long Baseline Array (VLBA) at twelve epochs (one observation per month from January to December 2011) at 15, 24, and 43 GHz. For the absolute orientation of the electric vector position angles (EVPA) we used the D-terms method; we also confirm its accuracy. We also used \u3b3-ray data from the Fermi Large Area Telescope on weekly time bins throughout 2011. Results: The source shows polarized emission, and its properties vary with time, frequency, and location along the jet. The core mean polarization fraction is generally between 1% and 2%, with a 4% peak at 43 GHz in March; the polarization angle is variable, mainly at 15 GHz, where it changes frequently, and less so at 43 GHz, where it oscillates in the range 114\ub0-173\ub0. The jet polarization properties are more stable, with a fractional polarization of around 16% and a polarization angle nearly perpendicular to the jet axis. The average flux and photon index at \u3b3-ray energies are (17.4 \ub1 0.5) 7 10-8 ph cm-2 s-1 and \u393 = 1.77 \ub1 0.02. The \u3b3-ray light curve shows variability, with a main peak of (38 \ub1 11) 7 10-8 ph cm-2 s-1 at the beginning of March and two later peaks centered on September 8 and November 13. The first \u3b3-ray peak appears to be associated with the peak in the core polarized emission at 43 GHz, as well as with the total intensity light curve. A discrete correlation function analysis yields a correlation coefficient of 0.54 at zero delay, with a significance level >99.7%. Conclusions: With this multifrequency study, we accurately determine the polarization properties of Mrk 421, both in the core and in the jet region. The radio and \u3b3-ray light curves are correlated. The observed EVPA variability at 15 GHz is partly due to opacity and partly to a variable Faraday rotation effect. To explain the residual variability of the intrinsic polarization angle and the low degree of polarization in the core region, we invoke a blend of variable cross-polarized subcomponents with different polarization properties within the beam