Linking Backlund and Monodromy Charges for Strings on AdS_5 x S^5

TL;DR

The paper establishes a covariant link between two central constructions of classical integrability for strings on AdS5 x S5: Backlund transformations and the monodromy (Lax) framework. By formulating a two-parameter Backlund transform and enforcing Virasoro compatibility, the authors derive a nonperturbative solution and show the Backlund generating functional equals a specific sum of monodromy quasi-momenta, with quasi-momenta positions fixed by the Backlund parameter. They derive a closed-form expression connecting Backlund and monodromy charges and confirm the relation in explicit rigid-string examples, supporting a covariant, gauge-independent picture of the integrable structure. The results have potential implications for covariant approaches to quantization and for understanding how classical integrable data encode the string spectrum on AdS5 x S5. These insights pave the way to describing quantum Backlund transformations and their relation to Bethe equations in a fully covariant setting, including extensions to the supersymmetric Green-Schwarz string.

Abstract

We find an explicit relation between the two known ways of generating an infinite set of local conserved charges for the string sigma model on AdS_5 x S^5: the Backlund and monodromy approaches. We start by constructing the two-parameter family of Backlund transformations for the string with an arbitrary world-sheet metric. We then show that only for a special value of one of the parameters the solutions generated by this transformation are compatible with the Virasoro constraints. By solving the Backlund equations in a non-perturbative fashion, we finally show that the generating functional of the Backlund conservation laws is equal to a certain sum of the quasi-momenta. The positions of the quasi-momenta in the complex spectral plane are uniquely determined by the real parameter of the Backlund transform.

Figures (1)

• Figure 1: The Bäcklund generating function $\mathbf{Q}(x)$ for $x\in [-1, 1]$ is given by the sum of the quasi-momenta in the upper and lower half-planes (semi-circles).