Looking for integrability on the worldsheet of confining strings

We study restrictions on scattering amplitudes on the worldvolume of branes and strings (such as confining flux tubes in QCD) implied by the target space Poincaré symmetry. We focus on exploring the conditions for the string worldsheet theory to be integrable. We prove that for a higher dimensional membrane the scattering amplitudes for the translational Goldstone modes (“branons”) are double soft. At one-loop double softness is generically violated for the string worldsheet scattering as a consequence of collinear singularities. Violation of double softness implies in turn the breakdown of integrability. We prove that if branons are the only gapless degrees of freedom then the worldsheet integrability is compatible with target space Poincaré symmetry only if the number of space-time dimensions is equal to D = 26 (a critical bosonic string), and for D = 3. We extend the analysis to include massless worldsheet fermions, resulting from spontaneous breakdown of the target space supersymmetry. We check that the tree-level integrability in this case is in one-to-one correspondence with the existence of a -symmetric Green-Schwarz (GS) action. As a byproduct we show that at the leading order in the derivative expansion an N = 1 $$ \mathcal{N}=1 $$ superstring without -symmetry in D = 3 , 4 , 6 , 10 dimensions exhibits an accidental enhanced supersymmetry and is equivalent to a -symmetric N = 2 $$ \mathcal{N}=2 $$ GS superstring