A derivation of Weyl gravity
N. Boulanger, M. Henneaux
TL;DR
The authors apply BRST-antifield cohomology to the problem of deriving Weyl gravity as a unique consistent deformation of linearized conformal gravity in four dimensions, under Lorentz invariance, a formal deformation parameter, and a bound of four derivatives. They systematically compute the relevant cohomology groups, establish that the invariant δ|d cohomology is obstructed only up to antifield number 2, and show that the first-order deformation is uniquely given by the Weyl vertex, with higher orders fixed to yield the full Weyl action. They also analyze theories with multiple Weyl gravitons, proving that cross-interactions are forbidden when the free limit is a sum of standard Weyl actions (positive-definite internal metric), but can arise if opposite-sign or indefinite internal metrics are allowed, with explicit constructions. The results extend to other spacetime dimensions and illuminate how the Bach tensor, Weyl tensor, and Cotton tensor organize the conformal-gauge structure in the cohomological classification. Overall, the paper demonstrates that Weyl gravity is uniquely determined by consistency requirements and clarifies the structure of possible multi-field generalizations at the level of deformations and gauge algebras.
Abstract
In this paper, two things are done. (i) Using cohomological techniques, we explore the consistent deformations of linearized conformal gravity in 4 dimensions. We show that the only possibility involving no more than 4 derivatives of the metric (i.e., terms of the form $\partial^4 g_{μν}$, $\partial^3 g_{μν} \partial g_{αβ}$, $\partial^2 g_{μν} \partial^2g_{αβ}$, $\partial^2 g_{μν} \partial g_{αβ} \partial g_{ρσ}$ or $\partial g_{μν} \partial g_{αβ} \partial g_{ρσ} \partial g_{γδ}$ with coefficients that involve undifferentiated metric components - or terms with less derivatives) is given by the Weyl action $\int d^4x \sqrt{-g} W_{\a\b\g\d} W^{\a\b\g\d}$, in much the same way as the Einstein-Hilbert action describes the only consistent manner to make a Pauli-Fierz massless spin-2 field self-interact with no more than 2 derivatives. No a priori requirement of invariance under diffeomorphisms is imposed: this follows automatically from consistency. (ii) We then turn to "multi-Weyl graviton" theories. We show the impossibility to introduce cross-interactions between the different types of Weyl gravitons if one requests that the action reduces, in the free limit, to a sum of linearized Weyl actions. However, if different free limits are authorized, cross-couplings become possible. An explicit example is given. We discuss also how the results extend to other spacetime dimensions.