Understanding GSW 2.2.3: Questions on Vertexoperators and Symmetric Traces

[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.

 

[R0man]

Hello René,

Thank you for sharing your questions on GSW 2.2.3 and vertex operators. I will do my best to provide some clarification on these topics.

Regarding the first question about the transformation law, it is important to note that the L_m's of the Virasoro algebra are not the same as the conserved charges mentioned on p. 65 and p. 72. The L_m's are generators of the conformal transformations, while the conserved charges are generators of the residual symmetries. Therefore, the transformation law tau -> tau -ie^{im tau} is a result of the conformal transformations generated by the L_m's, not the residual symmetries.

For your second question, the condition k_mu zeta^munu = 0 and tr zeta = 0 implies that the tensor zeta is traceless and satisfies the condition zeta^munu = zeta^numu. This means that the tensor is symmetric. To see this, you can use the definition of a symmetric tensor, which states that a tensor is symmetric if it is invariant under the exchange of its indices. In this case, exchanging the indices mu and nu in zeta^munu will not change the tensor since zeta^munu = zeta^numu. Therefore, under the given conditions, the tensor zeta is both traceless and symmetric.

I hope this helps clarify your questions. If you have any further inquiries, please feel free to ask.