Why are these open string oscillation modes identified with spin 1 particles on the brane?

Question

On page 55 of David Tong's String Theory lecture notes, during the discussion of the first excited states of open strings, two classes of states are identified, both of which are massless. The first class is described as,

Oscillators longitudinal to the brane, $$\alpha_{-1}^{a}|0 ; p\rangle \quad a=1, \ldots, p-1$$ The spacetime indices a lie within the brane so this state transforms under the $SO(1, p)$ Lorentz group. It is a spin 1 particle on the brane or, in other words, it is a photon. We introduce a gauge field $A_a$ with $a=0, \ldots, p$ lying on the brane whose quanta are identified with this photon.

It is not obvious to me that this state is a spin 1 particle. Why is it spin 1? Is that clear from the symmetry of the $SO(1,p)$ Lorentz group? Is this related to the Wigner classification? I think my not recognizing this might point to a gap in my QFT knowledge.

Answer 1

Te intuitive way to prove it is to notice that those open string oscillators have the same quantum numbers as a Yang-Mills gluon. See the answers to the question 2 coinciding D-branes leads to a U(2) gauge theory.

The idea is that both (first excited open string oscillators and YM gluons) have $d-2$ polarization states, come in $N^{2}$ varieties, two color/Chan-Paton degrees of freedom, same number of Lorentz indices (one) etc. So they are the same stuff.

The rigorous way to prove it is deriving the effective open string action and showing that those oscillators are dynamical and give rise to the Yang-Mills lagragian or by showing that the scattering amplitudes of those states reduces to Yang-Mills amplitudes in the so called "zero slope limit". Those approaches are worked in detail in the string theory textbook of Polchinski, Vol.1 , chapter 6, section 6.5.

The first approach should be enough to get convinced that string theory contains YM fields within it. However the second approach is always amazing because the first one leads open some possibilities (such as the presence of non-minimal terms in the effective action) that can be beautifully ruled out by the second approach.