There is intense interest in the B mode in CMB polarization: any primordial signal would be evidence of inflation. The informal talk will consist of two parts.

The first part will be a general introduction to CMB and its polarization, and the cosmological implications.

The second part of the talk is motivated by the need to understand precisely the common statement that the B mode is characterized by being “like a curl”. Specifically, we ask how a linear polarization field (a traceless rank-2 tensor) on a surface can be expressed in terms of scalar functions, thus providing an invariant representation and a separation into two components, one of which is the B mode. The case of a finite planar patch is sufficient to exhibit the key ideas, including the formal analogy with decomposing a vector field into a gradient plus a curl, with the B mode like the latter, but only in a formal sense. Generalization to curved (in particular spherical) surfaces is sketched, and global data are analyzed in terms of harmonic functions; this approach also provides a path to vector and tensor spherical harmonics.