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 tau(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 tau 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 tau in terms of modular functions of the UV coupling. 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. 
Marco Billo', Marialuisa Frau, Laurent Gallot and Alberto Lerda
hep-th