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

Abstract

We show that superconformal ${\cal N}=4,2$ algebras are well-suited to represent 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 realization corresponds naturally to standard smooth $\mathbb{R}^4$ the second describes the algebraic end of some small exotic smooth $\mathbb{R}^4$'s from the radial family 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 derives Witten-Reshetikhin-Turaev and Chern-Simons invariants of homology 3-spheres. These invariants are represented rather by false, quasi-modular, Ramanujan mock-type functions. Given the homology 3-spheres one determines exotic smooth structures of Freedman on $S^3\times \mathbb{R}$. In this way the fake ends are related 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 complexity of some separating from the infinity 3-dimensional submanifold. These separating manifolds can be chosen, in some exotic $\mathbb{R}^4$'s, to be homology 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 forms due to the noncommutativity of the codimension-one foliations of the homology 3-spheres. Then, the modification of modular forms is represented by the Connes-Moscovici construction ...