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We consider the local F-theory set-up corresponding to four D7 branes in typeI' theory, in which the exact axio-dilaton background tau(z) is identified withthe low-energy effective coupling of the four-dimensional N=2 super Yang-Millstheory with gauge group SU(2) and Nf=4 flavours living on a probe D3 braneplaced at position z. Recently, an intriguing relation has been found betweenthe correlators forming the chiral ring of the eight-dimensional theory on theD7 branes and the large-z expansion of the tau profile. Here we apply to theSU(2), Nf=4 theory some recursion techniques that allow to derive thecoefficients of the large-z expansion of tau in terms of modular functions ofthe UV coupling. In this way we obtain exact expressions for the elements ofthe eight-dimensional chiral ring that resum their instanton expansions,previously known only up to the first few orders by means of localizationtechniques.

The exact 8d chiral ring from 4d recursion relations