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We show that superconformal ${\cal N}=4,2$ algebras are well-suited torepresent some invariant constructions characterizing exotic $\mathbb{R}^4$relative to a given radial family. We examine the case of ${\cal N}=4,\hat{c}=4$ (at $r=1$ level) superconformal algebra which is realized on flat$\mathbb{R}^4$ and curved $S^3\times \mathbb{R}$. While the first realizationcorresponds naturally to standard smooth $\mathbb{R}^4$ the second describesthe algebraic end of some small exotic smooth $\mathbb{R}^4$'s from the radialfamily of DeMichelis-Freedman and represents the linear dilaton background$SU(2)_k\times \mathbb{R}_Q$ of superstring theory.  From the modular properties of the characters of the algebras one derivesWitten-Reshetikhin-Turaev and Chern-Simons invariants of homology 3-spheres.These invariants are represented rather by false, quasi-modular, Ramanujanmock-type functions. Given the homology 3-spheres one determines exotic smoothstructures of Freedman on $S^3\times \mathbb{R}$. In this way the fake ends arerelated to the SCA ${\cal N}=4$ characters.  The case of the ends of small exotic $\mathbb{R}^4$'s is more complicated.One estimates the complexity of exotic $\mathbb{R}^4$ by the minimal complexityof some separating from the infinity 3-dimensional submanifold. Theseseparating manifolds can be chosen, in some exotic $\mathbb{R}^4$'s, to behomology 3-spheres. The invariants of such homology 3-spheres are, again,obtained from the characters of SCA, ${\cal N}=4$.  Next we take into account the modification of the algebra of modular formsdue to the noncommutativity of the codimension-one foliations of the homology3-spheres. Then, the modification of modular forms is represented by theConnes-Moscovici construction ...

Towards superconformal and quasi-modular representation of exotic smooth R^4 from superstring theory I