Dimensional Reduction of Fermions in Brane Worlds of the Gross-Neveu Model

TL;DR

This paper investigates how fermions can be dimensionally reduced in a brane world scenario within the 3D Gross-Neveu model, revealing a mechanism for brane stabilization linked to baryon asymmetry and fermion localization.

Contribution

It provides an exact solution for a stable brane world with localized fermions and demonstrates a stabilization mechanism based on baryon asymmetry.

Findings

Fermions localize on domain walls and anti-walls, forming massive Dirac fermions.

Brane separation increases exponentially with correlation length, making the world effectively 2D.

Stability of the brane world requires extreme baryon asymmetry.

Abstract

We study the dimensional reduction of fermions, both in the symmetric and in the broken phase of the 3-d Gross-Neveu model at large N. In particular, in the broken phase we construct an exact solution for a stable brane world consisting of a domain wall and an anti-wall. A left-handed 2-d fermion localized on the domain wall and a right-handed fermion localized on the anti-wall communicate with each other through the 3-d bulk. In this way they are bound together to form a Dirac fermion of mass m. As a consequence of asymptotic freedom of the 2-d Gross-Neveu model, the 2-d correlation length \xi = 1/m increases exponentially with the brane separation. Hence, from the low-energy point of view of a 2-d observer, the separation of the branes appears very small and the world becomes indistinguishable from a 2-d space-time. Our toy model provides a mechanism for brane stabilization: branes made of fermions may be stable due to their baryon asymmetry. Ironically, our brane world is stable only if it has an extreme baryon asymmetry with all states in this ``world'' being completely filled.