In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree , just like ordinary spherical harmonics, but have an additional spin weight that reflects the additional symmetry. A special basis of harmonics can be derived from the Laplace spherical harmonics , and are typically denoted by , where and are the usual parameters familiar from the standard Laplace spherical harmonics. In this special basis, the spin-weighted spherical harmonics appear as actual functions, because the choice of a polar axis fixes the gauge ambiguity. The spin-weighted spherical harmonics can be obtained from the standard spherical harmonics by application of spin raising and lowering operators. In particular, the spin-weighted spherical harmonics of spin weight are simply the standard spherical harmonics: Spaces of spin-weighted spherical harmonics were first identified in connection with the representation theory of the Lorentz group . They were subsequently and independently rediscovered by and applied to describe gravitational radiation, and again by as so-called "monopole harmonics" in the study of Dirac monopoles. Spin-weighted functions Regard the sphere as embedded into the three-dimensional Euclidean space . At a point on the sphere, a positively oriented orthonormal basis of tangent vectors at is a pair of vectors such that where the first pair of equations states that and are tangent at , the second pair states that and are unit vectors, the penultimate equation that and are orthogonal, and the final equation that is a right-handed basis of . A spin-weight function is a function accepting as input a point of and a positively oriented orthonormal basis of tangent vectors at , such that for every rotation angle . Following , denote the collection of all spin-weight functions by . Concretely, these are understood as functions on } satisfying the following homogeneity law under complex scaling This makes sense provided is a half-integer. Abstractly, is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle of the Serre twist on the complex projective line . A section of the latter bundle is a function on } satisfying Given such a , we may produce a spin-weight function by multiplying by a suitable power of