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.
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