T-duality and hints of generalized geometry in string $α'$ corrections

Abstract

We examine the structure of higher-derivative string corrections under a cosmological reduction and make connection to generalized geometry and T-duality. We observe that, while the curvature R µ νρσ (Ω + ) of the generalized connection with torsion, Ω + = Ω + 1 2 H, is an important component in forming T-duality invariants, it is necessarily incomplete by itself. We revisit the tree-level α R 2 corrections to the bosonic and heterotic string in the language of generalized geometry and explicitly demonstrate that additional H-field couplings are needed to restore T-duality invariance. We also comment on the structure of the T-duality completion of tree-level α 3 R 4 in the type II string.

Higher-derivative couplings and torsional Riemann curvature

“…Apart from the couplings H 2 Q, the above results are exactly the couplings that have been found in [20] by the tree-level five-point functions including the correct normalization of the couplings in the structure H αβγ H µνρ Q αβγµνρ that have been clarified in [31]. The presence of these terms have been also confirmed by T-duality in [32,22]. Note that the five-point S-matrix calculations can not fix the Ricci and scalar curvatures, hence, one should also remove these terms in the expansion of ǫ 9 ǫ 9 H 2 R 3 term in [20].…”

“…It has been observed in [19] that there is no scheme in which the NS-NS couplings found in [18] can be written in terms of only the nonlinear generalized Riemann curvature. In fact, the sphere-level couplings of two B-fields and three gravitons at eight-derivative order have been found in [20] and shown that they can be written in terms of the generalized Riemann curvature and torsion H. In this paper, we are going to show that the metric, B-field and dilaton couplings at orders α ′2 , α ′3 that have been found in [21,18] by T-duality for the closed spacetime manifolds, can be written in a particular scheme in terms of only generalized Riemann curvature and H. The couplings at order α ′ have been written in terms of the generalized Riemann curvature and H in [22]. An outline of the paper is as follows: In section 2 we show that using the most general field redefinitions, Bianchi identities and adding total derivative terms, the effective action of the bosonic string theory at order α ′2 can be written in terms of the torsional Riemann curvature and H. In section 3, we repeat the same calculation for the NS-NS couplings of type II superstring theory at order α ′3 .…”

Higher-derivative couplings and torsional Riemann curvature

“…It has been observed in [19] that there is no scheme in which the NS-NS couplings found in [18] can be written in terms of only the nonlinear generalized Riemann curvature. In fact, the sphere-level couplings of two B-fields and three gravitons at eight-derivative order have been found in [20] and shown that they can be written in terms of the generalized Riemann curvature and torsion H. In this paper, we are going to show that the metric, B-field and dilaton couplings at orders α 2 , α 3 that have been found in [18,21] by T-duality for the closed spacetime manifolds, can be written in a particular scheme in terms of only generalized Riemann curvature and H. The couplings at order α have been written in terms of the generalized Riemann curvature and H in [22]. An outline of the paper is as follows: In section 2 we show that using the most general field redefinitions, Bianchi identities and adding total derivative terms, the effective action of the bosonic string theory at order α 2 can be written in terms of the torsional Riemann curvature and H. In section 3, we repeat the same calculation for the NS-NS couplings of type II superstring theory at order α 3 .…”

“…Using field redefinitions, Bianchi identities and total derivative terms, one can write the above action in terms of the generalized Riemann curvature and H. There are various schemes to write the above action in terms of the generalized Riemann curvature. The couplings in one particular scheme has been found in [22]…”

Completing R4 using O(d, d)

“…A cosmological reduction of all spatial dimensions has also been considered[12][13][14], but this is not enough to fix the form of the D-dimensional action.…”