The generalized Bergshoeff-de Roo identification. Part II

Abstract

We recently introduced a T-duality covariant mechanism to compute all-order higher-derivative interactions in the heterotic string. Here we extend the formalism to account for a two-parameter family of corrections that also include the bosonic string and HSZ theory. We use our result to compute the full second order Double Field Theory (DFT) for generic values of the parameters, including the generalized Green-Schwarz transformation and its invariant action.

O(D, D) and the string α′ expansion: an obstruction

“…We show that there is no R 3 invariant. Such terms are known to appear in the bosonic string effective action at order α 2 [23], but our findings are consistent with this since such terms will be generated as part of the R 2 invariant by requiring the corrected Lorentz transformations to close to higher orders in α [22].…”

“…This shows that no (independent) O(D, D) invariant R 3 -terms exist. Of course, R 3 -terms can be, and are, part of the higher order completion of the R 2 -invariant [22]. What we have shown here is that they cannot come with an independent coefficient.…”

“…We have found that the only higher derivative O(D, D) invariant up to this order is of the form Riemann squared. In fact it is known to capture the first α correction to the bosonic and heterotic string effective actions [15] and it seems likely that its completion accounts also for the α 2 correction in these theories [21,22]. The absence of any R 4 invariant at order α 3 is problematic however.…”

“…2 The corrections come about through a Green-Schwarz like modification of the double Lorentz transformations. This correction in fact induces a whole tower of α -corrections, by requiring the closure of the corrected transformations, and the resulting terms have the right form to be able to account also for the α 2 -corrections to the bosonic and heterotic string [21,22]. 3 Recently these results have been used to find the order α correction to generalizations of T-duality known as non-abelian and Poisson-Lie T-duality [24][25][26] and closely related integrable deformations of the string sigma model [27] (see also [28,29]).…”

Effective action of type II superstring theories at order α′3: NS-NS couplings

“…It has been shown in [26][27][28] that for d > 1 the Green-Schwarz type mechanism is required to have O(d, d, R) symmetry at four-derivative order. Another T-duality based framework for constructing the higher derivative couplings in string theory is the Double Field Theory [29][30][31][32][33][34][35][36][37][38] in which the dimension of spacetime is doubled and the O(D, D, R) symmetry is imposed on the 2D-dimensional theory before using dimensional reduction.…”