On Stabilization of Magnetically Charged Brane Shell and Over-extremality

In string theory, we can geometrically realize a metastable state by wrapping D5-branes and anti D5-branes to a singular manifold. We consider wrapping D3-branes to the internal space in this setup. These D3-branes dissolve into the domain wall, which interpolates true vacua and false vacua, forming a bound state. The remnant of the D3-branes can be seen as a background magnetic field on the domain wall, which appears to an observer in 4D spacetime as a magnetically charged, spherically symmetric shell. This brane shell has finite radii due to the nonlinearity peculiar to string theory, even at the probe level. We demonstrate a new stabilization mechanism of the brane shell in 4D spacetime. We add a general relativity-inspired gravitational correction to the brane shells and investigate the influence on its potential. As a result, the potential value at the horizon will be relatively larger than the potential minimum in parameter regions where the influence of gravity is large, and even non-perturbative instabilities can be removed. Moreover, we show the existence of over-extremal states such that $(gQ)^{2}\geq G_{4}m^{2}$ is satisfied in regions where the magnetic field is sufficiently large. At least in our model, this over-extremal shell cannot be completely stabilized by gravitational correction. This paper also addresses the dilemma between stabilizing and achieving an over-extremal state.