Virasoro group

Template:Short description In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by Vir)Template:Sfnp 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 $S^1$, whose points are labelled by a real coordinate $x$ subject to the identification $x\sim x+2\pi$, is a smooth map $f:ℝ\to ℝ:x\mapsto f(x)$ such that $f(x+2\pi )=f(x)+2\pi$ and $f^{\prime }(x)>0$. 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 $\widetilde{\text{Diff}}{}^{+}(S^1)$.

Definition

The Virasoro group is the universal central extension of $\widetilde{\text{Diff}}{}^{+}(S^1)$.Template:Sfnp The extension is defined by a specific two-cocycle, which is a real-valued function $𝖢(f,g)$ of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle: $$𝖢(f,g)\equiv -\frac{1}{48\pi }{\int }_0^{2\pi }\log [(f\circ g{)}^{\prime }(x)]\frac{g^{″}(x)\text{d}x}{g^{\prime }(x)}.$$ In these terms, the Virasoro group is the set $\widetilde{\text{Diff}}{}^{+}(S^1)\times ℝ$ of all pairs $(f,\alpha )$, where $f$ is a diffeomorphism and $\alpha$ is a real number, endowed with the binary operation $$(f,\alpha )\cdot (g,\beta )=(f\circ g,\alpha +\beta +𝖢(f,g)).$$ 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.Template:Sfnp 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.Template:Sfnp

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 $(\xi ,\alpha )$, where $\xi =\xi (x){\partial }_x$ is a vector field on the circle and $\alpha$ is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism $f(x)=x+ϵ\xi (x)$. The Lie bracket of pairs $(\xi ,\alpha )$ then follows from the multiplication defined above, and can be shown to satisfyTemplate:Sfnp

$[(\xi ,\alpha ),(\zeta ,\beta )]=([\xi ,\zeta ],-\frac{1}{24\pi }{\int }_0^{2\pi }\text{d}x\xi (x){\zeta }^{‴}(x))$

where the bracket of vector fields on the right-hand side is the usual one: $[\xi ,\zeta ]=(\xi (x){\zeta }^{\prime }(x)-\zeta (x){\xi }^{\prime }(x)){\partial }_x$. Upon defining the complex generators $$L_m\equiv (-i{e}^{imx}{\partial }_x,-\frac{i}{24}{\delta }_{m,0}),Z\equiv (0,-i),$$ the Lie bracket takes the standard textbook form of the Virasoro algebra:Template:Sfnp $$[L_m,L_n]=(m-n)L_{m+n}+\frac{Z}{12}m(m^2-1){\delta }_{m+n}.$$

The generator $Z$ 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 $Z$ is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.

Properties

Since each diffeomorphism $f$ must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function $f(x)-x$), 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.Template:Sfnp

References

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