From coadjoint orbits to scale invariant WZNW type actions and 2D quantum gravity action

Supergravitons interacting with the super-Virasoro group

“…Here we note that this is akin to the pairing of adjoint elements with coadjoint elements in the geometric actions [4,5,6]. This analogy follows since the conjugate momentum X ij0 transforms as an adjoint element (see Eqs.…”

“…In two dimensions it turns out that, one of the components of this rank two diffeomorphism tensor appears as the background quadratic differential present in computations of the two dimensional gravitational anomaly [3]. Its presence determines the symplectic structure on the coadjoint orbit of the Virasoro group [4,5] and is a signature that the diffeomorphism fields serve as classical gravitational fields in two dimensions. In other words the constant quadratic differentials b ⋆± that are often seen in the literature (for example [15]), are precisely the D ±± components of the diffeomorphism field in light-cone coordinates.…”

Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras

“…For the Virasoro algebra and also affine Lie algebras in one dimension these orbits can then be directly related to two dimensional field theories that are conformal field theories. These are the geometric actions [27,1,9]. Another reason for studying these representations comes when one studies the elements of the coadjoint representation and adjoint representation as conjugate variables of a field theory [20].…”

Superspace geometrical realization of the N-extended super Virasoro algebra and its dual

“…The coadjoint representation for infinite dimensional algebras [3,6] has appeared in the string theory literature for some time. Its uses include the study of chiral anomalies [10][11][12], geometric quantization of the Virasoro group [7], the study of orthogonal field theories [4,5,8] and recently in relation to AdS 3 quantum gravity [9].…”