The AdS3 x S3 x S3 x S1 Hernandez-Lopez Phases: a Semiclassical Derivation

TL;DR

This paper derives the Hernandez-Lopez phases for strings in AdS3 x S3 x S3 x S1 using semiclassical methods, accounting for modes absent in previous calculations and clarifying technical issues, with results matching known all-loop phases.

Contribution

It provides a semiclassical derivation of Hernandez-Lopez phases for AdS3 x S3 x S3 x S1, including modes and parameters previously unaccounted for, and clarifies technical subtleties.

Findings

Agreement with all-loop dressing phase in AdS3 x S3 x T4

Exact 1/2 agreement in AdS3 x S3 x S3 x S1 for any nd polarizations

Inclusion of modes absent in previous semiclassical calculations

Abstract

This note calculates the Hernandez-Lopez phases for strings in AdS3 x S3 x S3 x S1 by semiclassical methods using the d(2,1;\alpha)^2 algebraic curve. By working at general \alpha\ we include modes absent from previous semiclassical calculations of this phase in AdS3 x S3 x T4, and in particular can study the scattering of particles of different mass. By carefully re-deriving the semiclassical formula we clarify some issues of antisymmetrisation, cutoffs and surface terms which could safely be ignored in AdS5 x S5, and some issues about the terms c_1,s which were absent there. As a result we see agreement with the recently calculated all-loop dressing phase in the AdS3 x S3 x T4 case, and exactly 1/2 this in the general case AdS3 x S3 x S3 x S1 for any \alpha\ and any (light) polarisations.