Even Symplectic Supermanifolds and Double Field Theory

Abstract

Over many decades, the word “double” has appeared in various contexts, which at times seem to be unrelated.¹ Several have some relation to mathematical physics. Recently, this has become particularly striking in double field theory(DFT). 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 (Drinfel’d, Funktsional Anal Prilozhen 26(1):78–80, 1992) and of Lie bialgebras (Drinfel’d, Teoret Mat Fiz 95(2):226–227, 1993).

Quoting Voronov (Commun Math Phys 315(2): 279–310, 2012):

Double Lie algebroids arose in the works on double Lie groupoids (Mackenzie, Adv Math 94(2):180–239, 1992; Mackenzie, Adv Math 154(1):46–75, 2000a) and in connection with an analog for Lie bialgebroids of the classical Drinfel’d double of Lie bialgebras (Mackenzie, Adv Math 94(2):180–239, 1992; Mackenzie, Electron Res Announc Am Math Soc 4, 74–87, 1998)…. Suppose (A,A*) is a Lie bialgebroid over a base M…. Mackenzie in (Mackenzie, Adv Math 94(2):180–239, 1992; Mackenzie, Electron Res Announc Am Math Soc 4, 74–87, 1998; Mackenzie, Notions of double for Lie algebroids, arXiv:math/0011212, 2000b) and Roytenberg (Courant algebroids, derived brackets and even symplectic supermanifolds, 1999) suggested two different constructions based on the cotangent bundles T ^* A and ${T^*\Pi A}$, respectively. Here ${\Pi}$ 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 double field theory 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.