# Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory

## TL;DR

This paper explores the mathematical structure of gauge symmetries in double field theory, revealing that Vaisman algebroid arises from a Drinfel'd double of Lie algebroids and linking the strong constraint to Lie bialgebroid compatibility.

## Contribution

It demonstrates that Vaisman algebroid can be understood as a Drinfel'd double of Lie algebroids within the geometric framework of para-Hermitian manifolds, connecting gauge symmetry and algebraic structures.

## Key findings

- Vaisman algebroid is derived from a Drinfel'd double of Lie algebroids.
- The strong constraint in DFT is linked to Lie bialgebroid compatibility.
- Lagrangian subbundles serve as Dirac-like structures in the algebroid framework.

## Abstract

The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel'd double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian subbundles $(L,\tilde{L})$ in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for $(L,\tilde{L})$ be a Lie bialgebroid. The analysis provides a foundation toward the "coquecigrue problem" for the gauge symmetry in DFT.