The Kaluza-Klein Melvin Solution in M-theory

TL;DR

The paper analyzes the Kał uża-Klein Melvin background in M-theory and shows that IIA with magnetic field $B$ is dual to 0A with $B'=B-1/R$, connected by a 9-11 flip that also links to NS-NS Melvin fluxes. It derives the maximal field $|B|=1/R$ and demonstrates that spin structures determine inequivalent vacua within $-rac{1}{R}<B\,\le\,\frac{1}{R}$, with shifts $B\to B\pm\frac{1}{R}$ exchanging these structures. The study computes partition functions and spectra for NS-NS backgrounds, establishing invariance under $B\to B+2/R$ and relating the critical theory to a type IIA orbifold $S_1/(-1)^F\sigma_{1/2}$. Nonperturbative instabilities are analyzed via Kerr instantons: in IIA they manifest as D6/$\overline{D6}$ nucleation, while in 0A they appear as Witten's bubble expansion, with Euclidean actions providing decay rates. Overall, the work links Melvin backgrounds across M-theory reductions, clarifies decay channels, and highlights a deep connection between magnetic flux, spin structure, and nonperturbative string dynamics.

Abstract

We study some aspects of the Kaluza-Klein Melvin solution in M-theory. The associated magnetic field has a maximal critical value $B=\pm 1/R$ where $R$ is the radius of the compactification circle. It is argued that the Melvin background of type IIA with magnetic field $B$ and of type 0A with magnetic field $B'=B-1/R$ are equivalent. Evidence for this conjecture is provided using a further circle compactification and a `9-11' flip. We show that partition functions of nine-dimensional type IIA strings and of a $(-1)^Fσ_{1/2}$ type IIA orbifold both with NS-NS Melvin fluxtubes are related by such shift of the magnetic field. Then the instabilities of both IIA and 0A Melvin solutions are analyzed. For each theory there is an instanton associated to the decay of spacetime. In the IIA case the decay mode is associated to the nucleation of $D6/D\bar{6}$-brane pairs, while in the 0A case spacetime decays through Witten's bubble production.