# QFT T-duality, Massless vector states

1. Dec 10, 2017

### binbagsss

1. The problem statement, all variables and given/known data

2. Relevant equations

above,below

3. The attempt at a solution

so I think I understand the background of these expressions well enough, very briefly, changing the manifold from $R^n$ to a cylindrical one- $R^{(n-1)}^{+1}$ we need to cater for winding modes, the momentum and winding momentum for the circular dimension can not take arbitrary values and are quantified, $n,m \in Z$

And importantly, the level-matching constraint is no longer required to hold and instead replaced by the second equation in c) .

For the combinations I get:

a) $n=m=0$ $N=\bar{N}=1$
b) $n=m=1=N$ $\bar{N}=0$
c) $n=2$ $m=0=N=\bar{N}$
d) $m=2$ $n=N=\bar{N}=0$

I am completely stuck on which of these combinations transforms as a vector. The only notes relevant to it I seem to have is the following attached, (bit underlined in pink):

Is this referring to the ladder operator carrying a transverse index? or the state |p> ?

So out of the combinations above I have:

a) would require both a $\alpha^j$ and a $\bar{\alpha^j}$
b) would require just a $\alpha^j$
c) & d) would require no ladder operators.

Is the above relevant/needed at all or not, for what transforms as a vector or what doesn't, what defintion am I needing to go by here?

Last edited by a moderator: Dec 10, 2017
2. Dec 15, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Dec 19, 2017

### binbagsss

bump please, thank you so much.