O(D, D) and the string α′ expansion: an obstruction
Stanislav Hronek, Linus Wulff
Abstract
A bstract Double Field Theory (DFT) is an attempt to make the O ( d, d ) T-duality symmetry of string theory manifest, already before reducing on a d -torus. It is known that supergravity can be formulated in an O ( D, D ) covariant way, and remarkably this remains true to the first order in α′ . We set up a systematic way to analyze O ( D, D ) invariants, working order by order in fields, which we carry out up to order α′ 3 . At order α′ we recover the known Riemann squared invariant, while at order α′ 2 we find no independent invariant. This is compatible with the α′ expansion in string theory. However, at order α′ 3 we show that there is again no O ( D, D ) invariant, in contradiction to the fact that all string theories have quartic Riemann terms with coefficient proportional to ζ (3). We conclude that DFT and similar frameworks cannot capture the full α′ expansion in string theory.