  Recent theoretical work has suggested that Lyman-alpha nebulae could be
substantially polarized in the Lyman-alpha emission line, depending on the
geometry, kinematics, and powering mechanism at work. Polarization observations
can therefore provide a useful constraint on the source of ionization in these
systems. In this Letter, we present the first Lyman-alpha polarization
measurements for a giant Lyman-alpha nebula at z~2.656. We do not detect any
significant linear polarization of the Lyman-alpha emission:
P_{Lyman-alpha}=2.6+/-2.8% (corrected for statistical bias) within a single
large aperture. The current data also do not show evidence for the radial
polarization gradient predicted by some theoretical models. These results rule
out singly scattered Lyman-alpha (e.g., from the nearby AGN) and may be
inconsistent with some models of backscattering in a spherical outflow.
However, the effects of seeing, diminished signal-to-noise ratio, and angle
averaging within radial bins make it difficult to put strong constraints on the
radial polarization profile. The current constraints may be consistent with
higher density outflow models, spherically symmetric infall models,
photoionization by star formation within the nebula or the nearby AGN, resonant
scattering, or non-spherically symmetric cold accretion (i.e., along
filaments). Higher signal-to-noise ratio data probing to higher spatial
resolution will allow us to harness the full diagnostic power of polarization
observations in distinguishing between theoretical models of giant Lyman-alpha
nebulae.
