Quantum corrections to short folded superstring in AdS 3 × S 3 × M 4

Abstract

We consider integrable superstring theory on AdS 3 × S 3 × M 4 where M 4 = T 4 or M 4 = S 3 × S 1 with generic ratio of the radii of the two 3-spheres. We compute the oneloop energy of a short folded string spinning in AdS 3 and rotating in S 3 . The computation is performed by world-sheet small spin perturbation theory as well as by quantizing the classical algebraic curve characterizing the finite-gap equations. The two methods give equal results up to regularization contributions that are under control. One important byproduct of the calculation is the part of the energy which is due to the dressing phase in the Bethe Ansatz. Remarkably, this contribution E dressing 1 turns out to be independent on the radii ratio. In the M 4 = T 4 limit, we discuss how E dressing 1 relates to a recent proposal for the dressing phase tested in the su(2) sector. We point out some difficulties suggesting that quantization of the AdS 3 classical finite-gap equations could be subtler than the easier AdS 5 × S 5 case.

All-loop Bethe ansatz equations for AdS3/CFT2

“…The equations involve four undetermined scalar factors that play the role of a dressing phase. In contrast with what happens in AdS 5 /CFT 4 and AdS 4 /CFT 3 , it appears that crossing symmetry imposes that these factors differ from the BES phase [11], that seems to be consistent with the recent findings of [28,19]. Here we have investigated their form only in the large h limit (i.e., semi-classical strings), and established that they must also non-trivially couple nodes of different mass -another new feature with respect to [4,6].…”

“…However, taking the product of the two equations in (2.13) we find a crossing equation for S 15) where σ BES is the BES phase [11]. This form of the phase is similar to what was recently found in AdS 3 × S 3 × T 4 [19], as well as for Pohlmeyer reduced strings on AdS 3 × S 3 [20].…”

“…A significant one is the α → 0 limit of the theory, where one sphere blows up and two modes become massless. Upon compactification, this would give an AdS 3 ×S 3 ×T 4 background, that also appears to be integrable [4,5,19]. Understanding how these massless modes decouple may shed light on how to incorporate the AdS 3 × S 3 × S 3 × S 1 modes that the are not captured by the coset construction.…”

“…When h ≪ 1, let us expand 19) in the left sector, where u i are finite as h → 0, and similarly in the right sector. If we assume that the scalar factors S ij expand trivially in this limit, we indeed find that the Bethe ansatz takes the form (4.18).…”

Dressing phases ofAdS3/CFT2

“…Expanding on this, in [32] the exact integrable S-matrix and associated Bethe Ansatz of the integrable system associated to Type IIB string theory on AdS 3 ×S 3 ×T 4 with R-R flux was constructed. 3 A number of recent papers have performed important perturbative calculations for string theory on AdS 3 backgrounds [26,[34][35][36][37][38][39][40]. Further, the integrability of a family of AdS 3 × S 3 × T 4 theories with both R-R and NS-NS flux has been investigated [41].…”