  This is the second part of the work where quasi-modular forms emerge from
small exotic smooth $\mathbb{R}^4$'s grouped in a fixed radial family. SU(2)
Seiberg-Witten theory when formulated on exotic $\mathbb{R}^4$ from the radial
family, in special foliated topological limit can be described as SU(2)
Seiberg-Witten theory on flat standard $\mathbb{R}^4$ with the gravitational
corrections derived from coupling to ${\cal N}=2$ supergravity.
  Formally, quasi-modular expressions which follow the Connes-Moscovici
construction of the universal Godbillon-Vey class of the codimension-1
foliation, are related to topological correlation functions of superstring
theory compactified on special Callabi-Yau manifolds. These string correlation
functions, in turn, generate Seiberg-Witten prepotential and the couplings of
Seiberg-Witten theory to ${\cal N}=2$ supergravity sector. Exotic 4-spaces are
conjectured to serve as a link between supersymmetric and non-supersymmetric
Yang-Mills theories in dimension 4.
