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In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field whose (locally defined) flow defines conformal transformations, that is, preserve up to scale and preserve the conformal structure.
Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry. The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics. Thus the length of a tangent vector cannot be defined, but the angle between two ...
A plane: the locus of x is a plane if A = P, a vector with a zero n o component. In a homogeneous projective space such a vector P represents a vector on the plane n o =1 that would be infinitely far from the origin (ie infinitely far outside the null cone), so g(x).P =0 corresponds to x on a sphere of infinite radius, a plane. In particular:
Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with g ~ {\displaystyle {\tilde {g}}} , while those unmarked with such will be associated with g {\displaystyle g} .)
A conformal field theory is quasi-rational if the fusion product of two indecomposable representations is a sum of finitely many indecomposable representations. [9] For example, generalized minimal models are quasi-rational without being rational.
This Killing vector field generates the special conformal transformation. The colour indicates the magnitude of the vector field at that point. A toy example for a Killing vector field is on the upper half-plane M = R y > 0 2 {\displaystyle M=\mathbb {R} _{y>0}^{2}} equipped with the Poincaré metric g = y − 2 ( d x 2 + d y 2 ...
Several specific conformal groups are particularly important: The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V = ()
The Virasoro conformal blocks that are described in this article are associated to a certain type of representations of the Virasoro algebra: highest-weight representations, in other words Verma modules and their cosets. [2] Correlation functions that involve other types of representations give rise to other types of conformal blocks. For example: