The Odd story of $\alpha'$-corrections

The $\alpha'$-deformed frame-like Double Field Theory (DFT) is a T-duality and gauge invariant extension of DFT in which generalized Green-Schwarz transformations provide a gauge principle that fixes the higher-derivative corrections. It includes all the first order $\alpha'$-corrections of the bosonic and heterotic string low energy effective actions and of the Hohm-Siegel-Zwiebach $\alpha'$-geometry. Here we gauge this theory and parameterize it in terms of a frame, a two-form, a dilaton, gauge vectors and scalar fields. This leads to a unified framework that extends the previous construction by including all duality constrained interactions in generic (gauged/super)gravity effective field theories in arbitrary number of dimensions, to first order in $\alpha'$.

 Closed strings can wrap non-contractible cycles in space-time giving winding numbers that have no analogue in particle theory. Gauge invariance Beyond sugra: ' corrections  The string effective actions have higher-derivative corrections Bosonic string a = b =', Heterotic string a =  ', b=0, Type II a = b = 0 Can the gauge principle be deformed so that it requires and fixes the higher derivative corrections?

Green-Schwarz transformations
 The heterotic string has such a deformation: the Green-Schwarz transformation  This transformation requires and fixes the higher derivative corrections to the three-form field strength  However, the heterotic string contains in addition a Riemann squared term, which is not required by the GS transformations  Since T-duality mixes B and G we look for a duality covariant generalization of the GS transformation.
and the generalized frame and invariant dilaton as • The non-standard transformation of g  can be removed through a first order non-covariant field redefinition such that .
• Generically for B this is not possible and gives the GS transformation in the heterotic string (a = '

Scherk-Schwarzing Green-Schwarz
where which requires non-covariant field redefinitions such that and similarly for the other fields The generalized Green-Schwarz transformation implies      However, an accelerating space time solution cannot be obtained from only the lowest order terms in the supergravity action.
 The '-corrections to the four dimensional scalar potential, eventually combined with non-perturbative effects, offer the possibility not only to modify the Minkowski minima to a small value but also to stabilize some of the massless modes, modify the flat directions of the lowest order theory or change the slow roll behavior in inflationary models.

Four dimensional de Sitter solutions?
 Four dimensional maximally symmetric de Sitter solutions have been ruled out in the perturbative '-expansion of string theory from generic analysis of both the spacetime and the worldsheet theories  However, the '-corrections can be combined with non-perturbative quantum corrections or localized sources to produce solutions with properties that cannot be obtained from two-derivative supergravity.
 There are examples of AdS 4 solutions in type IIB or in the heterotic string, in which the leading order Minkowski ground states are broken by higher-derivative terms that generate a nonzero .
 We have studied the '-corrections to the known Minkowski vacua of N=4 gauged sugra in n=7, and found no modifications.

Summary and conclusions
 The traditional formulation of DFT has a duality covariant gauge symmetry principle based on a generalized Lie derivative that determines the two-derivative effective action uniquely.
 Different parametrizations allow to make contact with the standard universal bosonic sector of supergravity and lowerdimensional half-maximal gauged supergravities.
 The duality covariant gauge symmetry principle can be exended to include first-order deformations that account for the first order '-corrections to the bosonic and heterotic string effective actions and to half-maximal gauged sugra in arbitrary dimensions as well as to other duality covariant theories.

Future
• One remarkable aspect of the effective action is that the scalar potential receives an unambiguous first order correction.
• Understanding how this correction affects the vacuum structure may have interesting phenomenological consequences.