The modification of the energy spectrum of charged particles by exotic open 4-smoothness via superstring theory

In this paper we present a model where the modified Landau-like levels of charged particles in a magnetic field are determined due to the modified smoothness of ${\mathbb{R}}^4$ as underlying structure of the Minkowski spacetime. Then the standard smoothness of ${\mathbb{R}}^4$ is shifted to the exotic ${\mathbb{R}}_k^4$, $k=2p$, $p=1,\mathrm{2...}$. This is achieved by superstring theory using gravitational backreaction induced from a strong, almost constant magnetic field on standard ${\mathbb{R}}^4$. The exact string background containing flat ${\mathbb{R}}^4$ is replaced consistently by the curved geometry of $SU(2{)}_k\times \mathbb{R}$ as part of the modified exact backgrounds. This corresponds to the change of smoothness on ${\mathbb{R}}^4$ from the standard ${\mathbb{R}}^4$ to some exotic ${\mathbb{R}}_k^4$. The calculations of the spectra are using the CFT marginal deformations and Wess-Zumino-Witten (WZW) models. The marginal deformations capture the effects of the magnetic field as well as its gravitational backreactions. The spectra depend on even level $k$ of WZW on SU(2). At the same time the WZ term as element of $H^3(SU(2),\mathbb{R})$ determines also the exotic smooth ${\mathbb{R}}_k^4$. As the consequence we obtain a non-zero mass-gap emerges in the spectrum induced from the presence of an exotic ${\mathbb{R}}_k^4$.