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
An orientation-preserving diffeomorphism of the circle [math]\displaystyle{ S^1 }[/math], whose points are labelled by a real coordinate [math]\displaystyle{ x }[/math] subject to the identification [math]\displaystyle{ x\sim x+2\pi }[/math], is a smooth map [math]\displaystyle{ f:\mathbb{R}\to\mathbb{R}:x\mapsto f(x) }[/math] such that [math]\displaystyle{ f(x+2\pi)=f(x)+2\pi }[/math] and [math]\displaystyle{ f'(x)\gt 0 }[/math]. 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 [math]\displaystyle{ \widetilde{\text{Diff}}{}^+(S^1) }[/math].
Definition
The Virasoro group is the universal central extension of [math]\displaystyle{ \widetilde{\text{Diff}}{}^+(S^1) }[/math].[2]:sect. 4.4 The extension is defined by a specific two-cocycle, which is a real-valued function [math]\displaystyle{ \mathsf{C}(f,g) }[/math] of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle: [math]\displaystyle{ \mathsf{C}(f,g) \equiv -\frac{1}{48\pi}\int_0^{2\pi} \log\big[f'\big(g(x)\big)\big] \frac{g''(x)\,\text{d}x}{g'(x)}. }[/math] In these terms, the Virasoro group is the set [math]\displaystyle{ \widetilde{\text{Diff}}{}^+(S^1)\times\mathbb{R} }[/math] of all pairs [math]\displaystyle{ (f,\alpha) }[/math], where [math]\displaystyle{ f }[/math] is a diffeomorphism and [math]\displaystyle{ \alpha }[/math] is a real number, endowed with the binary operation [math]\displaystyle{ (f,\alpha)\cdot(g,\beta) = \big(f\circ g,\alpha+\beta+\mathsf{C}(f,g)\big). }[/math] 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.[2] 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.[2]
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 [math]\displaystyle{ (\xi,\alpha) }[/math], where [math]\displaystyle{ \xi=\xi(x)\partial_x }[/math] is a vector field on the circle and [math]\displaystyle{ \alpha }[/math] is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism [math]\displaystyle{ f(x)=x+\epsilon\xi(x) }[/math]. The Lie bracket of pairs [math]\displaystyle{ (\xi,\alpha) }[/math] then follows from the multiplication defined above, and can be shown to satisfy[3]:sect. 6.4 [math]\displaystyle{ \big[(\xi,\alpha),(\zeta,\beta)\big] = \bigg([\xi,\zeta],-\frac{1}{24\pi}\int_0^{2\pi}\text{d}x\,\xi(x)\zeta'''(x)\bigg) }[/math] where the bracket of vector fields on the right-hand side is the usual one: [math]\displaystyle{ [\xi,\zeta]=(\xi(x)\zeta'(x)-\zeta(x)\xi'(x))\partial_x }[/math]. Upon defining the complex generators [math]\displaystyle{ L_m\equiv\Big(-ie^{imx}\partial_x,-\frac{i}{24}\delta_{m,0}\Big), \qquad Z\equiv (0,-i), }[/math] the Lie bracket takes the standard textbook form of the Virasoro algebra:[4] [math]\displaystyle{ [L_m,L_n] = (m-n)L_{m+n}+\frac{Z}{12}m(m^2-1)\delta_{m+n}. }[/math]
The generator [math]\displaystyle{ Z }[/math] 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 [math]\displaystyle{ Z }[/math] is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.
Properties
Since each diffeomorphism [math]\displaystyle{ f }[/math] must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function [math]\displaystyle{ f(x)-x }[/math]), 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.[2]
References
- ↑ Bahns, Dorothea; Bauer, Wolfram; Witt, Ingo (2016-02-11) (in en). Quantization, PDEs, and Geometry: The Interplay of Analysis and Mathematical Physics. Birkhäuser. ISBN 978-3-319-22407-7. https://books.google.com/books?id=AouRCwAAQBAJ&pg=PA263.
- ↑ 2.0 2.1 2.2 2.3 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, doi:10.1007/978-3-319-61878-4, ISBN 978-3319618784
- ↑ 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
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