Is this possible to define the Virasoro algebra on the plane with both time and space infinite

Question

As we know, for 1+1D CFT, we have "radial quantization" to define the Virasoro algebra on the manifold with time or space to be periodic, like cylinder. But can we do that for both coordinates goes from​ \((-\infty,\infty)\)? I think if we do mode expansion as cylinder case, we would get the continuum Fourier integral.

I mean that on cylinder we can expand the energy-momentum tensor as\[T(w)=\sum_{n\in Z} L_n^{cyl}e^{-nw}\] ​where \(w=\tau-ix\)​. On cylinder, spacial coordinate \(x\)​ is periodic. But if we consider the case without this periodicity that both​ \(x\)​ and \(\tau\) does not perform as periodic coordinate and without any boundary, my naive expansion of the energy-momentum tensor would be something like\[T(w)=\int dk L(k)e^{-kw}\] And what do these expansion mode represent?

Comments

It looks as if you have written $infty$ instead of $\infty$ where the infinity symbol does not display correctly. For some reason I can not correct it myself in the editor.

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Edited. There was a missing backslash in front of "infty".

Answer 1

The Virasoro algebra is the Lie algebra of diffeomorphisms of the circle. These diffeomorphisms extend to the punctured plane as conformal transformations (actually there are two extensions, holomorphic and antiholomorphic), but most of the extensions have singularities at z = 0 or z = infinity. There are three generators that can be extended to the whole sphere which generate PSL(2,C), the group of Mobius transformations of the Riemann sphere. On the other hand, if we ask them to extend to the torus (periodic radial coordinate), then only one (complex) generator extends, corresponding to translation (modulus) and rotation (argument). Indeed the group of conformal transformations of the torus is just C* (sometimes plus some discrete piece for special conformal structures like a square torus).

Comments

Thanks. And I mean on cylinder we can expand the energy-momentum tensor as \[T(w)=\sum_{n\in Z} L_ne^{-nw}\]where \(w=\tau-ix\). On cylinder, spacial coordinate \(x\) is periodic, but if we consider the case without this periodicity that both \(x\) and \(\tau\) does not perform as periodic coordinate and without any boundary, my naive expansion of the energy-momentum tensor would be something like\[\int dk L(k) e^{-kw}\] And what do these expansion modes represent?

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It's best to think about T(z) as a holomorphic function on the punctured plane. Because of the $L_{-n}$'s there's a singularity at the origin, so there's no way to write it in usual Fourier series in the plane. Think of the expansion as Laurent series around the origin.