The all-loop integrable spin-chain for strings on AdS3 × S 3 × T 4: the massive sector

Abstract

We bootstrap the all-loop dynamic S-matrix for the homogeneous psu(1, 1|2) 2 spinchain believed to correspond to the discretization of the massive modes of string theory on AdS 3 × S 3 × T 4 . The S-matrix is the tensor product of two copies of the su(1|1) 2 invariant S-matrix constructed recently for the d(2, 1; α) 2 chain, and depends on two antisymmetric dressing phases. We write down the crossing equations that these phases have to satisfy. Furthermore, we present the corresponding Bethe Ansatz, which differs from the one previously conjectured, and discuss how our construction matches several recent perturbative calculations.

Massive S-matrix of AdS3×S3×T4 superstring theory with mixed 3

“…Then we can define the following tensor product states |y = |φ ⊗ |φ , |z = |ψ ⊗ |ψ , |ζ = |φ ⊗ |ψ , |χ = |ψ ⊗ |φ , (2.4) where |φ is bosonic and |ψ is fermionic. The factorization property means that the S-matrix for {y, z, ζ, χ} can be constructed from an S-matrix for {φ, ψ}, which takes the following form 5) where the signs ± represent the charges, i.e. correspond to the fields and their conjugates.…”

“…For the case of pure RR flux (q = 0) an all-loop conjecture for the S-matrix for the massive modes was made in [5] using symmetry algebra considerations and integrability constraints. 4…”

“…4 The phase factors still remain to be explicitly determined from crossing condition and the requirement of correspondence with string perturbation theory, see a discussion in [5].…”

“…Here we shall extend the tree-level (h → ∞) S-matrix (section 2) found in [1] from the superstring action to the exact in h result using symmetry algebra considerations [4,1] and generalizing the pure RR (q = 0) result of [5]. A key idea is that while the superstring symmetry group and the symmetry algebra of the S-matrix do not depend on q, the representation of the latter on particle states does (section 3).…”

“…the inverse of string tension. 1 In the q = 0 case [5] the exact S-matrix (written in terms of the Zhukovsky variables x ± ) is completely fixed, up to two phases, by its symmetry algebra and satisfies the Yang-Baxter equation without the need to use explicit form of the dispersion relation. To generalize to the q = 0 case we find a new set of Zhukovsky variables x ± ± (the ± subscripts correspond to positively/negatively charged states) that are consistent with the q-modified dispersion relation and in terms of which the representation parameters and the exact S-matrix take the same form as in the q = 0 case (section 5).…”