5-Dimensional Chern-Simons Gauge Theory on an Interval: Massive Spin-2 Theory from Symmetry Breaking via Boundary Conditions
In this note, we revisit the 4-dimensional theory of massive gravity through compactification of an extra dimension and geometric symmetry breaking. We dimensionally reduce the 5-dimensional topological Chern-Simons gauge theory of (anti) de Sitter group on an interval. We apply non-trivial boundary conditions at the endpoints to break all of the gauge symmetries. We identify different components of the gauge connection as invertible vierbein and spin-connection to interpret it as a gravitational theory. The effective field theory in four dimensions includes the dRGT potential terms and has a tower of Kaluza-Klein states without massless graviton in the spectrum. The UV cut of the theory is the Planck scale of the 5-dimensional gravity $l^{-1}$. If $\zeta$ is the scale of symmetry breaking and $L$ is the length of the interval, then the masses of the lightest graviton $m$ and the level $n$ (for $n<Ll^{-1}$) KK gravitons $m_{\rm KK}^{(n)}$ are determined as $m=(\zeta L^{-1})^{\frac{1}{2}}\ll m_{\rm KK}^{(n)}=nL^{-1}$. The 4-dimensional Planck mass is $m_{\rm Pl}\sim (Ll^{-3})^{\frac{1}{2}}$ and we find the hierarchy $\zeta< m< L^{-1}<l^{-1}<m_{\rm Pl}$.