Virasoro group

In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by "Vir") 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.
An orientation-preserving diffeomorphism of the circle formula_1, whose points are labelled by a real coordinate formula_2 subject to the identification formula_3, is a smooth map formula_4 such that formula_5 and formula_6. 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 formula_7.
Definition.
The Virasoro group is the universal central extension of formula_7. The extension is defined by a specific two-cocycle, which is a real-valued function formula_9 of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle:
formula_10
In these terms, the Virasoro group is the set formula_11 of all pairs formula_12, where formula_13 is a diffeomorphism and formula_14 is a real number, endowed with the binary operation
formula_15
This operation is an associative group operation. This extension is the only central extension of the universal cover of the group of circle diffeomorphisms, up to trivial extensions. The Virasoro group can also be defined without the use explicit coordinates or an explicit choice of cocycle to represent the central extension, via a description the universal cover of the group.
Virasoro algebra.
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 formula_16, where formula_17 is a vector field on the circle and formula_14 is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism formula_19. The Lie bracket of pairs formula_16 then follows from the multiplication defined above, and can be shown to satisfy
formula_21
where the bracket of vector fields on the right-hand side is the usual one: formula_22. Upon defining the complex generators
formula_23
the Lie bracket takes the standard textbook form of the Virasoro algebra:
formula_24
The generator formula_25 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 formula_25 is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.
Properties.
Since each diffeomorphism formula_13 must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function formula_28), the Virasoro group is infinite-dimensional.
Coadjoint representation.
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.