More on integrable structures of superstrings in AdS4 × CP 3 and AdS2 × S 2 × T 6 superbackgrounds

Abstract

In this paper we continue the study, initiated in [1,2], of the classical integrability of Green-Schwarz superstrings in AdS 4 × CP 3 and AdS 2 × S 2 × T 6 superbackgrounds whose spectrum contains non-supercoset worldsheet degrees of freedom corresponding to broken supersymmetries in the bulk. We derive an explicit expression, to all orders in the coset fermions and to second order in the non-coset fermions, which extends the supercoset Lax connection in these backgrounds with terms depending on the non-coset fermions. An important property of the obtained form of the Lax connection is that it is invariant under Z 4 -transformations of the superisometry generators and the spectral parameter. This demonstrates that the contribution of the non-coset fermions does not spoil the Z 4 -symmetry of the super-coset Lax connection which is of crucial importance for the application of Bethe-ansatz techniques. The expressions describing the AdS 4 × CP 3 and AdS 2 × S 2 × T 6 superstring sigma-models and their Lax connections have a very similar form. This is because their amount of target-space supersymmetries complement each other to 32 = 24+8, the maximal number of 10d type II supersymmetries. As a byproduct, this similarity has allowed us to obtain the form of the geometry of the complete type IIA AdS 2 × S 2 × T 6 superspace to all orders in the coset fermions and to the second order in the non-coset ones.

Near BMN dynamics of the AdS 3 × S 3 × S 3 × S 1 superstring

“…While both cubic and quartic contributions are rather involved, it is gratifying to see that the sum of the two simplifies. For the SU (2) × SU (2) sector, we see that the mixing sector exactly cancels out, and the total energy is just the sum of the two distinct SU (2) sectors 13 . Note that for the y 1 coordinate, the energy is, up to an overall sign, the same as the SU (2) sector of AdS 5 × S 5 [42].…”

“…In this case the kappa-symmetry gauge-fixing which reduces the GS string to the supercoset model becomes inconsistent when the string moves only in AdS 3 ⊂ AdS 3 × S 3 × S 3 × S 1 or in AdS 3 × S 3 ⊂ AdS 3 × S 3 × T 4 . However we expect that just as in AdS 4 × CP 3 and AdS 2 × S 2 × T 6 it should be possible to prove the classical integrability of the full GS string to quadratic order in fermions also in this case along the lines of [12,13], although we will not address this question in the present paper. The (super)isometries of AdS 3 ×S 3 ×S 3 form two copies of D(2, 1; α) which is an exceptional supergroup with a free parameter α ∈ [0, 1] [32]…”

“…First steps in this direction were taken in [12] where the classical integrability of the full Green-Schwarz string was demonstrated to quadratic order in fermions (the integrability was also shown to higher order in fermions in a truncated model), see also [13] (and for a slightly different approach see [14]). (A similar problem appears in AdS 2 × S 2 × T 6 except there the supercoset model never describes all the physical fermions due to the low amount of supersymmetry, nevertheless the integrability of the Green-Schwarz action has been shown to hold to quadratic order in fermions [15,13].) In [16] an analysis of the integrable structures of yet another duality, namely AdS 3 /CF T 2 , was initiated.…”

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

“…To compare this dispersion relation with the one we calculate in the BMN limit one needs to rescale the spin-chain momentum as p = (p 1 ∓ q)/h (the shift by q is needed in the comparison because we have defined the massive modes such that their quadratic action 1 The corresponding supergravity backgrounds preserve 8, 16 and 32 supersymmetries respectively. For the discussion of classical integrability see the original papers [2][3][4][5][6] or, for a general treatment for strings on symmetric spaces, see [7]. 2 The two-loop partition function for folded strings in AdS5 ×S 5 , related to the cusp anomalous dimension on the gauge theory side, has been computed in [23][24][25][26][27][28][29].…”

“…13) 6 The fact that only even orders appear is a special feature of the AdS × S × T backgrounds that make them particularly suited to a perturbative treatment. 7 Here ∂± = ∂0 ± ∂1.…”