Emergence of $SU(2)\times SU(2)$ at the self-dual point in bosonic string theory

I want to understand the derivation of the equations 8.3.11 in Polchinski Vol 1.

I can understand that at the self-dual point the Kaluza-Klein momentum index $n$, the winding number $w$, and the left and the right oscillator number $N$ and $\tilde{N}$ are related as,

$$(n+w)^2 + 4N = (n-w)^2 + 4\tilde{N} = 4$$

Now one can see that this has two "new" sets of massless states,

$$n = w = \pm1, N=0, \tilde{N}=1; n = -w = \pm 1, N=1, \tilde{N}=0$$

and

$$n= \pm 2, w=N=\tilde{N} =0; w = \pm 2, n=N=\tilde{N}=0$$

• But after that I don't get the argument as to how the 4 states in the first of the above set is represented by vertex operators

$$:\bar{\partial}X^\mu e^{ik.X}\exp[\pm 2i\alpha '^{-1/2}X_L^{25}]:$$

and

$$:\partial X^\mu e^{ik.X}\exp[\pm 2i\alpha '^{-1/2}X_R^{25}]:$$

What is the derivation of the above?

• Also now looking at these 4+4 states how does the argument for existence of a $SU(2)\times SU(2)$ symmetry follow? The argument in the paragraph below 8.3.11 is hardly clear to me.
-

The correspondence between these vertex operators and the special massless states at the self dual radius can be understood as follows:

1) The coefficient of the exponent implies that they have the right winding and Kaluza-Klein quantum numbers.

2) They are spin-1 and they are massless.

The main point is that on the noncompactified dimensions they are either right or left movers (either $N$ $\tilde{N}$ are zero), therefore, they belong to the open-string sector. As can be seen, these vertex operators are just the open-string photon vertex operators. However, for a general compactification radius, these states become massive, only at the self dual radius, they become massless again.

As explained by Polchinski, the Kaluza-Klein charges of these massless vector Bosons are nonvanishing, thus they must belong to a non Abelian gauge multiplet. To verify that the symmetry is $SU(2)$ one has to actually compute the commutators.

Also, the mechanism that they acquire mass on the departure from the self dual radius is a stringy Higgs mechanism, where they absorb the coset generators of the broken symmetry. This mechanism is explained by Polchinski in the next section.

-