S-matrix on effective string and compactified membrane

Expanding Nambu-Goto action near infinitely long string vacuum one can compute scattering amplitudes of 2d massless fields representing transverse string coordinates. As was shown in (Dubovsky et al 2012 J. High Energy Phys. JHEP09(2012)044), the resulting S-matrix is integrable (provided appropriate local counterterms are added), in agreement with known free string spectrum and also with an interpretation of the static-gauge NG action as a TT ˉ deformation of a free massless theory. We consider a generalization of this computation to the case of a membrane, expanding its 3d action near an infinite membrane vacuum that has cylindrical RxS1 shape (we refer to such membrane as 'compactified'). Representing 3d fields as Fourier series in S 1 coordinate we get an effective 2d model in which the massless string modes are coupled to an infinite KK tower of massive 2d modes. We find that the resulting 2d S-matrix is not integrable already at the tree level. We also compute 1-loop scattering amplitude of massless string modes with all compactified membrane modes propagating in the loop. The result is UV finite and is a non-trivial function of the kinematic variables. In the large momentum limit or when the radius of S 1 is taken to infinity we recover the expression for the 1-loop scattering amplitude of the uncompactified R2 membrane. We also consider a 2d model which is the TT ˉ deformation to the free theory with the same massless plus infinite massive tower of modes. The corresponding 2d S-matrix is found, as expected, to be integrable. Contribution to the special issue of Journal of Physics A: 'Fields, Gravity, Strings and Beyond: In Memory of Stanley Deser'