Bethe Ansatz for Quantum-Deformed Strings

TL;DR

This paper compares two integrable quantum deformations of strings on AdS5×S5, showing they share the same Bethe equations and conserved charges, indicating a fundamental equivalence in their integrable structures.

Contribution

It demonstrates that two different η-deformations lead to identical Bethe equations and conserved charges, revealing a unified integrable structure.

Findings

Both deformations yield the same Bethe equations.

Eigenvalues of transfer matrices coincide for both deformations.

The asymptotic spectrum is described by the derived Bethe-Yang equations.

Abstract

Two distinct $\eta$-deformations of strings on AdS$_5\times$S$^5$ can be defined; both amount to integrable quantum deformations of the string non-linear sigma model, but only one is itself a superstring background. In this paper we compare their conjectured all-loop worldsheet S matrices and derive the corresponding Bethe equations. We find that, while the S matrices are apparently different, they lead to the same Bethe equations. Moreover, in either case the eigenvalues of the transfer matrix, which encode the conserved charges of each system, also coincide. We conclude that the integrable structure underlying the two constructions is essentially the same. Finally, we write down the full Bethe-Yang equations describing the asymptotic spectrum of the superstring background.