General string cosmologies at order α′3

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

“…Focusing on the cosmological reduction on T 9 , gauge invariant higher derivative invariants can be constructed out of Φ and N ± and their time derivatives. However, as shown in [49,56], the use of field redefinitions and on-shell equations of motion can eliminate all time derivatives of Φ and N ± , leading to a minimal basis consisting of traces of strings of N + and N − . In this case, the derivative counting is straightforward, as each N ± counts precisely one time derivative.…”

“…Following [50,56], it is convenient to choose a canonical basis that eliminates all powers of Tr(L 2 ) so that the basis becomes minimal when the H-field is truncated out (i.e. when M → 0).…”

Duality invariant string beta functions at two loops

“…This new method can be extended to higher loops, and the details of the three-loop case will be presented in a separate paper. Of course, the complications grow sharply with the number of loops, but it is quite conceivable that with a suitable automation of the computation one could eventually push it beyond the α 3 corrections that so far are the state-of-the-art [35,36]. The real goal, however, of determining the α -complete equations, whose general form is known thanks to the classification in [32], is still out of reach.…”

“…This procedure will be explained in section 3 by revisiting the one-loop computation and then be used in section 4 to determine the O(α ) coefficient in the cosmological classification of [32], thereby providing JHEP02(2022)109 an independent check of [33,34], where this coefficient was computed by direct dimensional reduction. Remarkably, the beta functions of the cosmological Polyakov action lead to the duality invariant beta functions (and hence field equations) rather directly, which is in contrast to the dimensional reduction procedure that requires some elaborate field redefinitions [33][34][35][36]. At one and two loops it is easy to see diagrammatically that potential duality violating terms (that would have to be removed by field redefinitions) do not arise.…”

Completing $R^4$ 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.2 Curiously, while the complicated (H ∧ H)R 3 and H 2 ∇H 2 R terms found above are required at tree-level by O(d, d), they are absent at one loop[16]. The one-loop R 4 -terms therefore seem to have a much simpler structure than the tree-level ones, even though in the type IIB case the purely metric terms are exactly the same.…”