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.

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: $${\displaystyle {\mathsf {C}}(f,g)\equiv -{\frac {1}{48\pi }}\int _{0}^{2\pi }\log {\big [}f'{\big (}g(x){\big )}{\big ]}{\frac {g''(x)\,{\text{d}}x}{g'(x)}}.}$$ 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.

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]