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,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,L) be a Lie bialgebroid. The analysis provides a foundation toward the "coquecigrue problem" for the gauge symmetry in DFT.

We continue to study doubled aspects of algebroid structures equipped with the Cbracket in double field theory (DFT). We find that a family of algebroids, the Vaisman (metric or pre-DFT), the pre-and the ante-Courant algebroids are constructed by the analogue of the Drinfel'd double of Lie algebroid pairs. We examine geometric implementations of these algebroids in the para-Hermitian manifold, which is a realization of the doubled space-time in DFT. We show that the strong constraint in DFT is necessary to realize the doubled and non-trivial Poisson structures but can be relaxed for some algebroids. The doubled structures of twisted brackets and those associated with group manifolds are briefly discussed. a h.mori(at)sci.kitasato-u.ac.jp b shin-s(at)kitasato-u.ac.jp

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The Vaisman algebroid is a kind of algebroid structure. It is de ned by an extension of the Courant algebroid, and phisically related to the gauge symmetry in Double Field Theory (DFT), which is an effective theory of string theory. DFT has T-duality as a manifest symmetry. In this study, we focus on the \doubled structure” in the Vaisman algebroid. It is already well known that some kind of Lie algebras are obtained by the Drinfel’d double of Lie bialgebras. The Courant algebroid is obtaind by the Drinfel’d double of Lie bialgebroids. We nd that the Vaisman algebroid can be obtained by an analogue of the \Drinfel’d double” of Lie algebroids. We discuss the algebraic origin of the strong constraint in DFT.