Symmetries in four-dimensional multi-spin-two field theory. Relations to Chern-Simons gravity and further applications

This thesis is devoted to the investigation of the symmetry in theories for spin-2 fields. In particular, we address the question whether the fully nonlinear theories for massless and massive spin-2 particles - namely standard general relativity and bimetric gravity, can be obtained from a pure gauge theory formulation. Furthermore, we explore the discrete symmetries that can arise in a theory for many metrics, when we choose one metric to describe the dynamics of spacetime while keeping all others exactly on an equal footing. To this end, we consider the Chern-Simons gauge theory in five dimensions valued in the anti-de Sitter group AdS${}_{4+1}$ = SO(4, 2). Using the fact that this group is isomorphic to the conformal group in four dimensions, C${}_{3+1}$, we express the theory in the basis for the conformal algebra. The gauge fields corresponding to the translations P${}_a$ and special conformal transformations K${}_a$, denoted by e${}^a$ and ${\iota }^a$, are then interpreted as two vierbeine after implementing several dimensional reduction schemes. At the four dimensional level we find for different schemes that the Chern-Simons theory reduces to a generalisation of standard general relativity and first-order conformal gravity. Moreover, we introduce a doubled Chern-Simons theory in five dimensions with symmetry group SO(4, 2) x SO(4, 2). We break down the symmetry to SO(3, 1) x SO(2) and after a dimensional reduction, we recover a generalisation of 4-dimensional bimetric gravity. In all cases, we discuss the residual symmetry of the actions at the 4-dimensional level. We also consider a ghost-free multigravity theory with the physical metric g${}_{\mu \nu }$ and the satellite metrics $f_{\mu \nu }^{\left(p\right)}$ where p = 1,...,N. For this, we explore the discrete symmetry that arises when the interchange between the satellite metrics leaves the action invariant and also when both square roots $\pm$$\sqrt{g^{-1f^{\left(p\right)}}}$ in the action lead to the same interaction potential between the physical and the satellite metrics. The global symmetry group that arises is isomorphic to S${}_N$ x ($Z_2$)${}^N$. Moreover, we analyse the mass spectrum of the theory with the discrete symmetry. We focus then in the trimetric case since the multimetric case behaves similarly and does not lead to new phenomenology. The massless spin-2 mode G${}_{\mu \nu }$ mediates the long-range gravitational force of the spacetime on which the massive spin-2 modes M${}_{\mu \nu }$ and ${\chi }_{\mu \nu }$ propagate. By computing the perturbative expansion, we analyse the vertices of the theory in terms of the spin-2 modes. We find that the spin-2 particle with the smaller mass, ${\chi }_{\mu \nu }$, is stable and in particular, it cannot decay into massless gravitons. We postulate that this spin-2 field can be a component of dark matter.