Classical and quantum integrability in AdS 2/CFT 1

Abstract

We investigate the type IIA string on AdS 2 × S 2 × T 6 supported by RR-flux which describes the gravitational side of the AdS 2 /CF T 1 correspondence. While the fourdimensional part AdS 2 × S 2 can be realized as a supercoset, the full superstring has both coset and non-coset excitations, the latter giving rise to massless worldsheet modes, a somewhat novel feature in AdS/CF T . The string is nevertheless known to be integrable at the classical level. In this paper we perform several computations checking aspects of both classical and quantum string integrability. At the classical level we compute energies for the near BMN string and successfully match these against Bethe ansatz predictions. Furthermore, integrability dictates a magnon dispersion relation which we compare with the poles of loop corrected propagators, at both the one and two-loop level. At one loop, where only tadpole diagrams contribute, we find that the bosonic and fermionic contributions sum up to zero. Under the assumption of worldsheet supersymmetry, we then compute the two-loop sunset diagram in the near flat space limit. As in AdS 5 × S 5 we find that the result fits

Integrable S-matrices, massive and massless modes and the AdS 2 × S 2 superstring

“…Furthermore, they may depend on the choice of Type II background [5][6][7][8][9] -in the decompactified light-cone gauge-fixed theory the T 6 formally has an SO(6) symmetry, however, this will be broken by the presence of RamondRamond fluxes. Initial investigations in this direction for a particular Type IIA background were carried out in [27,28], in which case the SO(6) is broken to U(3). However, different backgrounds related by T-duality will naively lead to different subgroups [5][6][7][8][9].…”

“…Defining the effective string tension 27) the tree-level S-matrix for the scattering of massive modes in the light-cone gauge AdS 2 × S 2 × T 6 superstring following from near-BMN perturbation theory can be be found by 7 We would like to thank the referee for suggesting the consideration of double crossing.…”

“…Furthermore, as a consequence of the form of the symmetry algebra and the integrability of the theory [21,27] we expect that the S-matrix for y, z, ζ and χ can be constructed as a graded tensor product of an S-matrix for φ and ψ, with each factor S-matrix invariant under the symmetry psu(1|1) R 3 . In this section we will construct the relevant massive representation of psu(1|1) R 3 .…”

The AdS n × S n × T 10−2n BMN string at two loops

“…Therefore calculations were done either in the so-called Near-Flat-Space limit [10][11][12][13][14][15], using generalized unitarity [16][17][18] or by computing only quantities that were explicitly finite [4,9,19,20]. Thus one could side-step the issue of regularization.…”

AdS 3/CFT 2, finite-gap equations and massless modes

“…On more general cosets however, the GRC may lead to non-trivial corrections to the residue conditions used in the literature. For example we expect such effects to arise in the AdS 2 × S 2 × S 2 × T 4 theories [65][66][67][68][69].…”