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Virasoro group - Wikipedia

  • ️Thu Feb 11 2016

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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.

An orientation-preserving diffeomorphism of the circle {\displaystyle S^{1}}, whose points are labelled by a real coordinate {\displaystyle x} subject to the identification {\displaystyle x\sim x+2\pi }, is a smooth map {\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto f(x)} such that {\displaystyle f(x+2\pi )=f(x)+2\pi } and {\displaystyle f'(x)>0}. 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 {\displaystyle {\widetilde {\text{Diff}}}{}^{+}(S^{1})}.

The Virasoro group is the universal central extension of {\displaystyle {\widetilde {\text{Diff}}}{}^{+}(S^{1})}.[2]: sect. 4.4  The extension is defined by a specific two-cocycle, which is a real-valued function {\displaystyle {\mathsf {C}}(f,g)} of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle: {\displaystyle {\mathsf {C}}(f,g)\equiv -{\frac {1}{48\pi }}\int _{0}^{2\pi }\log {\big [}(f\circ g)'(x){\big ]}{\frac {g''(x)\,{\text{d}}x}{g'(x)}}.} In these terms, the Virasoro group is the set {\displaystyle {\widetilde {\text{Diff}}}{}^{+}(S^{1})\times \mathbb {R} } of all pairs {\displaystyle (f,\alpha )}, where {\displaystyle f} is a diffeomorphism and {\displaystyle \alpha } is a real number, endowed with the binary operation {\displaystyle (f,\alpha )\cdot (g,\beta )={\big (}f\circ g,\alpha +\beta +{\mathsf {C}}(f,g){\big )}.} 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]

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 {\displaystyle (\xi ,\alpha )}, where {\displaystyle \xi =\xi (x)\partial _{x}} is a vector field on the circle and {\displaystyle \alpha } is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism {\displaystyle f(x)=x+\epsilon \xi (x)}. The Lie bracket of pairs {\displaystyle (\xi ,\alpha )} then follows from the multiplication defined above, and can be shown to satisfy[3]: sect. 6.4  {\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 )}} where the bracket of vector fields on the right-hand side is the usual one: {\displaystyle [\xi ,\zeta ]=(\xi (x)\zeta '(x)-\zeta (x)\xi '(x))\partial _{x}}. Upon defining the complex generators {\displaystyle L_{m}\equiv {\Big (}-ie^{imx}\partial _{x},-{\frac {i}{24}}\delta _{m,0}{\Big )},\qquad Z\equiv (0,-i),} the Lie bracket takes the standard textbook form of the Virasoro algebra:[4] {\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {Z}{12}}m(m^{2}-1)\delta _{m+n}.}

The generator {\displaystyle Z} 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 {\displaystyle Z} is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.

Since each diffeomorphism {\displaystyle f} must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function {\displaystyle f(x)-x}), the Virasoro group is infinite-dimensional.

Coadjoint representation

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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]

  1. ^ 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.
  2. ^ a b c d Guieu, Laurent; Roger, Claude (2007), L'algèbre et le groupe de Virasoro, Montréal: Centre de Recherches Mathématiques, ISBN 978-2921120449
  3. ^ 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
  4. ^ 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