Jacobi-Lie T-plurality

TL;DR

This paper introduces a new algebraic structure called DD$^+$, establishes its relation to Jacobi--Lie bialgebras, and demonstrates that Jacobi-Lie T-plurality is a symmetry in double field theory, with various examples provided.

Contribution

It generalizes the Drinfel'd double to DD$^+$, linking it to Jacobi--Lie bialgebras, and proposes Jacobi-Lie T-plurality as a new symmetry in double field theory.

Findings

Established a one-to-one correspondence between DD$^+$ and Jacobi--Lie bialgebras.

Constructed generalized frame fields satisfying specific algebraic relations.

Demonstrated Jacobi-Lie T-plurality as a symmetry of double field theory with examples.

Abstract

We propose a Leibniz algebra, to be called DD$^+$, which is a generalization of the Drinfel'd double. We find that there is a one-to-one correspondence between a DD$^+$ and a Jacobi--Lie bialgebra, extending the known correspondence between a Lie bialgebra and a Drinfel'd double. We then construct generalized frame fields $E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+$ satisfying the algebra $\mathcal{L}_{E_A}E_B = - X_{AB}{}^C\,E_C\,$, where $X_{AB}{}^C$ are the structure constants of the DD$^+$ and $\mathcal{L}$ is the generalized Lie derivative in double field theory. Using the generalized frame fields, we propose the Jacobi-Lie T-plurality and show that it is a symmetry of double field theory. We present several examples of the Jacobi-Lie T-plurality with or without Ramond-Ramond fields and the spectator fields.