Virasoro group
In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by Vir)[1] is an infinite-dimensional Lie group defined as the universal central extension of the group of diffeomorphisms of the circle. The corresponding Lie algebra is the Virasoro algebra, which has a key role in conformal field theory (CFT) and string theory.
The group is named after Miguel Ángel Virasoro and Raoul Bott.
Background[edit]
An orientation-preserving diffeomorphism of the circle , whose points are labelled by a real coordinate subject to the identification , is a smooth map such that and . The set of all such maps spans a group, with multiplication given by the composition of diffeomorphisms. This group is the universal cover of the group of orientation-preserving diffeomorphisms of the circle, denoted as .
Definition[edit]
The Virasoro group is the universal central extension of .[2]: sect. 4.4 The extension is defined by a specific two-cocycle, which is a real-valued function of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle:
Virasoro algebra[edit]
The Lie algebra of the Virasoro group is the Virasoro algebra. As a vector space, the Lie algebra of the Virasoro group consists of pairs , where is a vector field on the circle and is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism . The Lie bracket of pairs then follows from the multiplication defined above, and can be shown to satisfy[3]: sect. 6.4
The generator commutes with the whole algebra. Since its presence is due to a central extension, it is subject to a superselection rule which guarantees that, in any physical system having Virasoro symmetry, the operator representing is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.
Properties[edit]
Since each diffeomorphism must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function ), the Virasoro group is infinite-dimensional.
Coadjoint representation[edit]
The Lie bracket of the Virasoro algebra can be viewed as a differential of the adjoint representation of the Virasoro group. Its dual, the coadjoint representation of the Virasoro group, provides the transformation law of a CFT stress tensor under conformal transformations. From this perspective, the Schwarzian derivative in this transformation law emerges as a consequence of the Bott–Thurston cocycle; in fact, the Schwarzian is the so-called Souriau cocycle (referring to Jean-Marie Souriau) associated with the Bott–Thurston cocycle.[2]
References[edit]
- ^ Bahns, Dorothea; Bauer, Wolfram; Witt, Ingo (2016-02-11). Quantization, PDEs, and Geometry: The Interplay of Analysis and Mathematical Physics. Birkhäuser. ISBN 978-3-319-22407-7.
- ^ a b c d Guieu, Laurent; Roger, Claude (2007), L'algèbre et le groupe de Virasoro, Montréal: Centre de Recherches Mathématiques, ISBN 978-2921120449
- ^ Oblak, Blagoje (2016), BMS Particles in Three Dimensions, Springer Theses, Springer Theses, arXiv:1610.08526, doi:10.1007/978-3-319-61878-4, ISBN 978-3319618784, S2CID 119321869
- ^ Di Francesco, P.; Mathieu, P.; Sénéchal, D. (1997), Conformal Field Theory, New York: Springer Verlag, doi:10.1007/978-1-4612-2256-9, ISBN 9780387947853