Minimal solution of the AdS/CFT crossing equation

TL;DR

The paper tackles the AdS/CFT crossing equation for the scalar factor $\sigma[u,v]$ in the integrable S-matrix and derives the minimal solution under generalanalytic assumptions. By working in mirror kinematics with the Jukowsky map $x[u]$, decomposing the dressing factor into $\sigma_1[x,v]$ and $\sigma_2[x,y]$, and solving a Riemann-Hilbert problem for $\chi[x,y]$, it reproduces the BES/BHL dressing phase in the minimal scenario and clarifies the possible CDD deformations. It further analyzes the analytic structure across multiple Riemann sheets, showing how the solution can be expressed via a gamma-function ratio and a dressing kernel, and relates the CDD freedom to meromorphic functions on the elliptic torus with a fixed periodicity. Overall, the work provides a constructive derivation of the BES/BHL phase, delineates the space of allowed CDD factors, and elucidates the phase’s torus-structured analytic properties relevant for AdS/CFT integrability.

Abstract

We solve explicitly the crossing equation under sufficiently general assumptions on the structure of the dressing phase. We obtain the BES/BHL dressing phase as a minimal solution of the crossing equation and identify the possible CDD factors.

Figures (2)

• Figure 1: The crossing $A^*$ and the mirror points $B$ in the $u$-plane.
• Figure 2: Analytical structure of $\sigma_2[x,y]$ as a function of $u$.