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Old Dec 28th 2017, 09:08 AM   #1
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QFT T-duality, Massless vector states

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

Part C) Please:

(attachment 1)

2. Relevant equations


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):

(attachment 2)

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.
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QFT T-duality, Massless vector states-pf2-1-.png   QFT T-duality, Massless vector states-pf2-2-.png  
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