The value $R=\alpha^{\prime 1/2}$ is the self-dual radius under T-duality. One may indeed extract the massless spectrum – the spectrum of all fields much lighter than $\alpha^{\prime -1/2}$.

Because the CFT has an $SU(2)\times SU(2)$ symmetry, as can be seen from the OPEs of the currents, the spacetime physics has this symmetry, too. Because one finds (spacetime) Lorentz vector states in the adjoint of $SU(2)\times SU(2)$, it is clear that this group is the gauge symmetry of the spacetime physics.

And indeed, one may verify that the tree-level scattering amplitudes for all the relevant string modes agree with the scattering amplitudes extracted for quanta of fields in the effective action that is (a bit schematically, especially when it comes to the parts unrelated to the enhanced gauge symmetry)
$$ S =\int d^{25}x\,\exp(2\phi) [R + (\partial_{[\lambda} B_{\mu\nu]})^2 + (\partial_\mu \phi)^2 -\frac 14 {\rm Tr}(F_{\mu\nu}F^{\mu\nu}) ] $$
So it is a 25-dimensional action because we ignore the 1 compactified dimension whose radius is stringy (dimensional reduction). In this 25-dimensional spacetime, there is the dilaton, the metric, the B-field, and an $SU(2)\times SU(2)$ gauge field, and they have more or less the expected terms in the effective action.

See Polchinski's Volume 1 from page 242 to 250+ or so. The effective action is probably not written there explicitly. However, you may find the 26D effective action for the uncompactified bosonic string theory on the top of page 114, reduce the dimension, and add the $SU(2)\times SU(2)$ Yang-Mills field, more or less getting the exact answer. The "Cartan" $U(1)\times U(1)$ part of the Yang-Mills action comes from the Kaluza-Klein $U(1)$ symmetry of the circle and from the components of the B-field $B_{\mu,25}$. This is "enhanced" by the extra "accidentally massless" states to the non-Abelian group.

Between equations 3.7.15 and 3.7.20 or so, Polchinski takes a different but ultimately equivalent strategy to derive the spacetime action. He derives the equations of motion from the requirement of the conformal symmetry on the world sheet (vanishing beta-functions etc.) and verifies that the same equations follow as the Euler-Lagrange equations from the spacetime action he "guesses" and "refines".

This post imported from StackExchange Physics at 2014-03-05 14:56 (UCT), posted by SE-user Luboš Motl