The exact 8d chiral ring from 4d recursion relations

We consider the local F-theory set-up corresponding to four D7 branes in type I' theory, in which the exact axio-dilaton background τ (z) is identified with the low-energy effective coupling of the four-dimensional N = 2 super Yang-Mills theory with gauge group SU(2) and Nf = 4 flavours living on a probe D3 brane placed at position z. Recently, an intriguing relation has been found between the correlators forming the chiral ring of the eight-dimensional theory on the D7 branes and the large-z expansion of the τ profile. Here we apply to the SU(2) Nf = 4 theory some recursion techniques that allow to derive the coefficients of the large-z expansion of τ in terms of modular functions of the UV coupling τ. In this way we obtain exact expressions for the elements of the eight-dimensional chiral ring that resum their instanton expansions, previously known only up to the first few orders by means of localization techniques.