Semiclassical spectrum of a Jordanian deformation of $AdS_5 \times S^5$

TL;DR

This paper analyzes a Jordanian deformation of the $AdS_5 imes S^5$ superstring, exploring its integrability, spectral curve, and semi-classical quantization, revealing that certain spectral properties are preserved at one-loop level.

Contribution

It introduces a detailed analysis of the semiclassical spectrum of a Jordanian deformed superstring, including the construction of the spectral curve and one-loop energy corrections.

Findings

Deformed model can be reformulated with twisted boundary conditions.

One-loop correction to energy matches undeformed case for specific solutions.

Unimodular and non-unimodular deformations share the same spectrum at one-loop.

Abstract

We study a Jordanian deformation of the $AdS_5 \times S^5$ superstring that preserves 12 superisometries. It is an example of homogeneous Yang-Baxter deformations, a class that generalises TsT deformations to the non-abelian case. Many of the attractive features of TsT carry over to this more general class, from the possibility of generating new supergravity solutions to the preservation of worldsheet integrability. In this paper, we exploit the fact that the deformed $\sigma$-model with periodic boundary conditions can be reformulated as an undeformed one with twisted boundary conditions, to discuss the construction of the classical spectral curve and its semi-classical quantisation. First, we find global coordinates for the deformed background, and identify the global time corresponding to the energy that should be computed in the spectral problem. Using the curve of the twisted model, we obtain the one-loop correction to the energy of a particular solution, and we find that the charge encoding the twisted boundary conditions does not receive an anomalous correction. Finally, we give evidence suggesting that the unimodular version of the deformation (giving rise to a supergravity background) and the non-unimodular one (whose background does not solve the supergravity equations) have the same spectrum at least to one-loop.