Metric Projective Geometry, BGG Detour Complexes and Partially Massless Gauge Theories

Abstract

Abstract. A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of projective geometry combines with structures known as Yang-Mills detour complexes to produce a general tool for generating invariant pseudo-Riemannian gauge theories. This produces (detour) complexes of differential operators corresponding to gauge invariances and dynamics. We show, as an application, that curved versions of these sequences give geometric characterizations of the obstructions to propagation of higher spins in Einstein spaces. Further, we show that projective BGG detour complexes generate both gauge invariances and gauge invariant constraint systems for partially massless models: the input for this machinery is a projectively invariant gauge operator corresponding to the first operator of a certain BGG sequence. We also connect this technology to the log-radial reduction method and extend the latter to Einstein backgrounds.

Massless and massive higher spins from anti-de Sitter space waveguide

“…It turns out that 1 Formally, the compactifications [9,10] and its (A)dS counterparts [7,8] may be viewed as projectively reducing on a conformal hypersurface. This viewpoint was further studied for partially massless spin-two system in [11]. Starting from non-unitary conformal gravity, this work showed that projective reduction yields partially massless spin-two system, which is also non-unitary.…”

“…10 In the path integral formulation, this amounts to choosing that the integration contour purely imaginary. 11 Here, we define the mass-squared equal to the mass-squared in flat space limit. Therefore, it differs from the mass-squared dictated by the Fierz-Pauli equations.…”

Dark energy from dynamical projective connections

“…The field equations for the metric and connection can be examined independently [42][43][44][45] in the context of Gauss-Bonnet, and because we are in four-dimensions, metric compatibility is still a solution to the field equations. Also, we have already mentioned in section 3 that when one of the members of a projective equivalence class [Γ a bc ] is a metric compatible connection, the projective Schouten tensor collapses to a constant times the metric [40,41]. D ab is that projective Schouten tensor when it is not dynamical.…”

Dynamical projective curvature in gravitation

“…[3.13] we recognizes D ab as precisely the projective Schouten tensor and Eq. [3.14] as λ times the projective Cotton-York tensor, [32][33][34]. From the projective curvature tensor, we calculate the components of the projective Ricci tensor K αβ ≡ K ρ αβρ to be,…”

“…This metric coincides with metric projective tractors at a certain scale [34,39,40,33,32,30]. To guarantee finite volume when λ is integrated from 0 to ∞ in the action functionals, we choose f (λ) so that…”