Questions on GSW 2.2.3

[Rene Meyer]

Hello,

I stumbled about two things in GSW section 2.2.3 on Vertexoperators,
that I don't really understand.

The first one is GSW's statement just before 2.2.54, p. 88, that the
L_m's of the Virasoro algebra generate transformations like

tau -> tau -ie^{im tau}

.... From what was said on p. 65 of the generators of the residual
symmetry and on p. 72 from the Virasoro generators I know that these
should be conserved charges, thus generating some transformations with
f(sigma^+) = e^imsigma^+ and f(sigma^-) = e^imsigma^-, which is at
sigma = 0 just e^imtau. But how to get from this result to the above
transformation law?

The second one is the statement on p. 92 that for the two conditions
k_mu zeta^munu = 0 and tr zeta = 0 the tensor zeta should be a
symmetric traceless tensor. Tracelessness is clear, but how to show
that under this condition the tensor should be symmetric?

I hope that these questions are not too elementary, but as I am new
with the string stuff, many elementary things bother me most,
sometimes.

René.

--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China

[Urs Schreiber]

On Wed, 31 Mar 2004, Rene Meyer wrote:

> The first one is GSW's statement just before 2.2.54, p. 88, that the
> L_m's of the Virasoro algebra generate transformations like
>
> tau -> tau -ie^{im tau}

To see this just go the other way round. Check that a field of conformal
weight J has the commutator given in (2.2.54).

> The second one is the statement on p. 92 that for the two conditions
> k_mu zeta^munu = 0 and tr zeta = 0 the tensor zeta should be a
> symmetric traceless tensor. Tracelessness is clear, but how to show
> that under this condition the tensor should be symmetric?

Symmetry follows implicitly from the form of formula (2.2.66), where
antisymmetric components of zeta do not contribute. k_m zeta^mn = 0
is responsible for the statement "... to be the polarization tensor..."
because this removes the unphysical timelike polarization.