Jacobi-Lie Models and Supergravity Equations

Poisson-Lie T-duality/plurality was recently generalized to Jacobi-Lie T-plurality formulated in terms of Double Field Theory and based on Leibniz algebras given by structure coefficients $f_{ab}{}^{c},f_{c}{}^{ab},$ and $Z_a,Z^a$. We investigate three- and four-dimensional sigma models corresponding to six-dimensional Leibniz algebras with $f_b{}^{ba}\neq 0$, $Z^a=0$. We show that these algebras are plural one to another and, moreover, to an algebra with $f_b{}^{ba}= 0$, $Z^a=0$. These pluralities are used for construction of Jacobi-Lie models. It was conjectured that plural models should satisfy Generalized Supergravity Equations. We have found examples of models satisfying ``true'' Generalized Supergravity Equations where no trivialization to usual Supergravity Equations is possible. On the other hand, we show that there are also models corresponding to algebras with $f_b{}^{ba}\neq 0$, $Z^a=0$ where the Killing vector appearing in Generalized Supergravity Equations either vanishes or can be removed by suitable gauge transformation. Such models then satisfy usual Supergravity Equations, i.e. vanishing beta function equations.