Even symplectic supermanifolds and double field theory

Andreas Deser and Jim Stasheff
arXiv:1406.3601 [math-ph]

Over many decades, the word “double” has appeared in various contexts, at times seemingly unrelated. Several have some relation to mathematical physics. Recently, this has become particularly strking in DFT (double field theory). Two ‘doubles’ that are particularly relevant are double vector bundles and Drinfel’d doubles. The original Drinfel’d double occurred in the contexts of quantum groups and of Lie bialgebras. Quoting T. Voronov: “Double Lie algebroids arose in the works on double Lie groupoids and in connection with an analog for Lie bialgebroids of the classical Drinfel’d double of Lie bialgebras…Suppose (A,A) is a Lie bialgebroid over a base M… Mackenzie and Roytenberg suggested two different constructions based on the cotangent bundles TA and TΠA, respectively. Here Π is the fibre-wise parity reversal functor.” Although the approaches of Roytenberg and of Mackenzie look very different, Voronov establishes their equivalence. We have found Roytenberg’s version to be quite congenial with our attempt to interpret the gauge algebra of DFT in terms of Poisson brackets on a suitable generalized Drinfel’d double. This double of a Lie bialgebroid (A,A) provides a framework to describe the differentials of A and A on an equal footing as Hamiltonian functions on an even symplectic supermanifold. A special choice of momenta explicates the double coordinates of DFT and shows their relation to the strong constraint determining the physical fields of double field theory.