Abstract
We dimensionally reduce the spacetime action of bosonic string theory, and that of the bosonic sector of heterotic string theory after truncating the Yang-Mills gauge fields, on a $d$-dimensional torus including all higher-derivative corrections to first order in $\alpha'$. A systematic procedure is developed that brings this action into a minimal form in which all fields except the metric carry only first order derivatives. This action is shown to be invariant under ${\rm O}(d,d,\mathbb{R})$ transformations that acquire $\alpha'$-corrections through a Green-Schwarz type mechanism. We prove that, up to a global pre-factor, the first order $\alpha'$-corrections are uniquely determined by ${\rm O}(d,d,\mathbb{R})$ invariance.
Highlights
String theory features the T-duality property according to which there is a nonlinear group action of Oðd; d; ZÞ on d-dimensional toroidal backgrounds such that all backgrounds in one orbit are physically equivalent
We dimensionally reduce the spacetime action of bosonic string theory, and that of the bosonic sector of heterotic string theory after truncating the Yang-Mills gauge fields, on a d-dimensional torus including all higher-derivative corrections to first order in α0
For the two-derivative effective action this symmetry was first shown explicitly for the reduction to one dimension by Veneziano and Meissner in Refs. [1,2] and later generalized to arbitrary d by Maharana and Schwarz [3]. It was proven by Sen, using closed string field theory, that the Oðd; d; RÞ symmetry of dimensionally reduced theories is present to all orders in α0 [4], but it remains as a highly nontrivial problem to display this symmetry when higher-derivative α0-corrections are included
Summary
String theory features the T-duality property according to which there is a nonlinear group action of Oðd; d; ZÞ on d-dimensional toroidal backgrounds such that all backgrounds in one orbit are physically equivalent. In this paper we complete the existing literature by giving the complete dimensionally reduced action for bosonic string theory to first order in α0, i.e., including all four-derivative terms, and prove its Oðd; d; RÞ invariance, presenting results that have recently been announced in Ref. To our surprise we find that there is no choice of field variables so that the full dimensionally reduced action can be written in terms of familiar Oðd; d; RÞ covariant variables (such as the generalized metric); rather, a generalized Green-Schwarz mechanism is required under which the (external) singlet B-field acquires nontrivial transformations under Oðd; d; RÞ, implying that the Oðd; d; RÞ action gets α0deformed This effect has been invisible in all truncations investigated so far, but it does mimic the situation in double field theory before reduction.
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