We present further mathematical results on a function appearing in the conformal blocks of four-point correlation functions with arbitrary primary operators. The H-function was introduced in a previous article and it has several interesting properties. We prove explicitly the recurrence relation as well as the D<sub>6</sub>-invariance presented previously. We also demonstrate the proper action of the differential operator used to construct the H-function.
We study the non-linear stability of fixed-point solutions to the α′-exact equations from O(d, d) invariant cosmology, with and without matter perturbations. Previous non-linear analysis in the literature is revisited, and its compatibility with known linear perturbation results is shown. Some formal aspects of cosmological perturbations in duality invariant cosmology are discussed, and we show the existence of time-reparameterization invariant variables for perturbations.
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