Ghost equations and diffeomorphism-invariant theories

Abstract

Four-dimensional Einstein gravity in the Palatini first order formalism is shown to possess a vector supersymmetry of the same type as found in the topological theories for Yang-Mills fields. A peculiar feature of the gravitationel theory, characterized by diffeomorphism invariance, is a direct link of vector supersymmetry with the field equation of motion for the Faddeev-Popov ghost of diffeomorphisms.

On the symmetries of BF models and their relation with gravity

“…For three dimensions and specific gauge groups, the BF model can be related directly to gravity. Thus, we could show that VSUSY still holds, as expected [11], within the gravity framework. However, VSUSY then turns out to be excluded from the BRST symmetry corresponding to the invariances with respect to diffeomorphisms and local Lorentz transformations.…”

“…Since the second commutator in the last line vanishes, we obtain If we pass over from the BF model to gravity by virtue of the identifications (2.47), we recover the VSUSY-variations (1.8) of gravity, i.e. the result of reference [11], now generalized by the presence of the antifields.…”

“…The action (1.6) can be gauge-fixed by considering a background metric and by choosing the Landau gauge. The gauge-fixed action is then invariant under VSUSYtransformations parametrized by a Killing vector field τ = τ µ ∂ µ [11]: the latter only act non-trivially on the ghost field ξ according to…”

“…The more general validity of VSUSY invariance in our approach is due to its inclusion in the Slavnov-Taylor identity whereas it is left outside in the other schemes. As for the usual gravitational formulation discussed in section 2.3, one is precisely in the latter situation since we were forced to keep VSUSY outside of the BRST operator: VSUSY then represents a rigid symmetry, v being a Killing vector field with respect to a background metric [11] (assuming again that the manifold under consideration admits such vector fields at all).…”

Symmetries and observables in topological gravity

“…Since ϕ describes a local shift of the diffeomorphism ghost, it parametrizes vector supersymmetry transformations [36]. By expanding equations (2.8) and (2.6) with respect to the ghost-number, we get the S-variations of all fields as well as the relation φ = i ϕ e. The latter is equivalent to the relation ϕ µ = φ a e µ a if we assume the vielbein to be invertible.…”