Do Drinfeld twists of $AdS_5 \times S^5$ survive light-cone quantization?

Submission summary

 As Contributors: Yannik Zimmermann Arxiv Link: https://arxiv.org/abs/2112.10279v2 (pdf) Date accepted: 2022-04-26 Date submitted: 2022-04-13 16:32 Submitted by: Zimmermann, Yannik Submitted to: SciPost Physics Core Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

Abstract

We study how a wide class of Abelian Yang-Baxter deformations of the AdS$_\mathsf{5} \times$S$^\mathsf{5}$ string behave at the quantum level. These deformations are equivalent to TsT transformations and conjectured to be dual to beta, dipole, and noncommutative deformations of SYM. Classically they correspond to Drinfeld twists of the original theory. To verify this expectation at the quantum level we compute and match (1) the bosonic two-body tree-level worldsheet scattering matrix of these deformations in the uniform light-cone gauge, and (2) the Bethe equations of the equivalent model with twisted boundary conditions. We find that for a generalization of gamma deformations of the BMN string the we are able to express the S matrix either through a Drinfeld twist or a shift of momenta. For deformations of the GKP string around the null-cusp solution we encounter calculational obstacles that prevent us from calculating the scattering matrix.

Published as SciPost Phys. Core 5, 028 (2022)

List of changes

We have addressed the points raised in the first report. Regarding 1. we have added a clarifying comment elaborating on the configuration (state) dependence of the boundary conditions in the text above equation (2.6). Regarding 2. we have rephrased the text to make it clear that in this section we were deriving a set of equations to be contrasted against other results later. Regarding 3., we agree that moving the deformation into boundary conditions is not inherently undesirable. However, it simply does not align with our goal to compute S matrices in the deformed model directly. We extended the sentence to clarify our intended meaning. Regarding 4., we agree that this is a subtle and relevant point. We have added a footnote that we believe quantum integrability to be essentially unaffected: the momentum shift in the Lagrangian should not affect the presence of sufficiently many conserved charges, but these charges would pick up particle species dependent momentum shifts, which in turn should lead to the shifted Yang-Baxter equation.

The second referee commented that it is not clear whether our problems surrounding deformations of the GKP string could be easily overcome. We agree that it is desirable to definitively settle this question. However, lack of a resolution is typically not easy to prove. As described, we have attempted various redefinitions and modifications without success. Moreover, the field redefinitions done in the undeformed setting, certainly blur the action of the conserved charges involved in the deformation, on the scattering fields of the Lagrangian. This suggests a hypothetical resolution would require more than e.g. deforming the classical solution we expand around.