QFT Tduality, Massless vector states
1. The problem statement, all variables and given/known data
Part C) Please on first attachment
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 $R^{(n1)}^{+1}$ 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 levelmatching 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):
2nd attachment
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?
Many thanks in advance
