  "Ly$\alpha$ nebulae" are giant ($\sim$100 kpc), glowing gas clouds in the
distant universe. The origin of their extended Ly$\alpha$ emission remains a
mystery. Some models posit that Ly$\alpha$ emission is produced when the cloud
is photoionized by UV emission from embedded or nearby sources, while others
suggest that the Ly$\alpha$ photons originate from an embedded galaxy or AGN
and are then resonantly scattered by the cloud. At least in the latter
scenario, the observed Ly$\alpha$ emission will be polarized. To test these
possibilities, we are conducting imaging polarimetric observations of seven
Ly$\alpha$ nebulae. Here we present our results for LABd05, a cloud at $z$ =
2.656 with an obscured, embedded AGN to the northeast of the peak of Ly$\alpha$
emission. We detect significant polarization. The highest polarization
fractions $P$ are $\sim$10-20% at $\sim$20-40 kpc southeast of the Ly$\alpha$
peak, away from the AGN. The lowest $P$, including upper-limits, are $\sim$5%
and lie between the Ly$\alpha$ peak and AGN. In other words, the polarization
map is lopsided, with $P$ increasing from the Ly$\alpha$ peak to the southeast.
The measured polarization angles $\theta$ are oriented northeast, roughly
perpendicular to the $P$ gradient. This unique polarization pattern suggests
that 1) the spatially-offset AGN is photoionizing nearby gas and 2) escaping
Ly$\alpha$ photons are scattered by the nebula at larger radii and into our
sightline, producing tangentially-oriented, radially-increasing polarization
away from the photoionized region. Finally we conclude that the interplay
between the gas density and ionization profiles produces the observed central
peak in the Ly$\alpha$ emission. This also implies that the structure of LABd05
is more complex than assumed by current theoretical spherical or cylindrical
models.
