Poisson-Lie duals of the $\eta$-deformed $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ superstring

TL;DR

This paper explores Poisson-Lie duals of the $ ext{AdS}_2 imes ext{S}^2 imes ext{T}^6$ superstring with $ ext{eta}$-deformation, analyzing their supergravity solutions and deriving explicit backgrounds, especially for the bosonic subalgebra case.

Contribution

It provides the first explicit construction of Poisson-Lie dual backgrounds for the $ ext{eta}$-deformed superstring, including detailed RR flux and dilaton analysis.

Findings

Metrics and B-fields match an analytic continuation of the $ ext{lambda}$-deformed model.

Explicit background derived for the bosonic subalgebra case.

Agreement with known $ ext{lambda}$-model embeddings in supergravity.

Abstract

We investigate Poisson-Lie duals of the $\eta$-deformed $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ superstring. The $\eta$-deformed background satisfies a generalisation of the type II supergravity equations. We discuss three Poisson-Lie duals, with respect to (i) the full $\mathfrak{psu}(1,1|2)$ superalgebra, (ii) the full bosonic subalgebra and (iii) the Cartan subalgebra, for which the corresponding backgrounds are expected to satisfy the standard type II supergravity equations. The metrics and B-fields for the first two cases are the same and given by an analytic continuation of the $\lambda$-deformed model on $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ with the torus undeformed. However, the RR fluxes and dilaton will differ. Focusing on the second case we explicitly derive the background and show agreement with an analytic continuation of a known embedding of the $\lambda$-deformed model on $\mathrm{AdS}_2 \times \mathrm{S}^2$ in type II supergravity.