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Register a new userA note about a pure spin-connection formulation of General Relativity and spin-2 duality in (A)dS
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We investigate the problem of finding a pure spin-connection formulation of General Relativity with non-vanishing cosmological constant. We first revisit the problem at the linearised level and find that the pure spin-connection, quadratic Lagrangian, takes a form reminiscent to Weyl gravity, given by the square of a Weyl-like tensor. Upon Hodge dualisation, we show that the dual gauge field in (A)dS$_D$ transforms under $GL(D)$ in the same representation as a massive graviton in the flat spacetime of the same dimension. We give a detailed proof that the physical degrees of freedom indeed correspond to a massless graviton propagating around the (anti-) de Sitter background and finally speculate about a possible nonlinear pure-connection theory dual to General Relativity with cosmological constant.
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Starting from the dual Lagrangians recently obtained for (partially) massless spin-2 fields in the Stueckelberg formulation, we write the equations of motion for (partially) massless gravitons in (A)dS in the form of twisted-duality relations. In both cases, the latter admit a smooth flat limit. In the massless case, this limit reproduces the gravitational twisted-duality relations previously known for Minkowski spacetime. In the partially-massless case, our twisted-duality relations preserve the number of degrees of freedom in the flat limit, in the sense that they split into a decoupled pair of dualities for spin-1 and spin-2 fields. Our results apply to spacetimes of any dimension greater than three. In four dimensions, the twisted-duality relations for partially massless fields that appeared in the literature are recovered by gauging away the Stueckelberg field.
General Relativity can be reformulated as a diffeomorphism invariant gauge theory of the Lorentz group, with Lagrangian of the type $f(Fwedge F)$, where $F$ is the curvature 2-form of the spin connection. A theory from this class with a generic $f$ is known to propagate eight degrees of freedom: a massless graviton, a massive graviton and a scalar. General Relativity in this formalism avoids extra degrees of freedom because the function $f$ is special and leads to the appearance of six extra primary constraints on the phase space variables. Our main new result is that there are other theories of the type $f(Fwedge F)$ that lead to six extra primary constraints. However, only in the case of GR the dynamics is such that these six primary constraints get supplemented by six secondary constraints, which gives the end result of two propagating degrees of freedom. This is how uniqueness of GR manifests itself in this ``pure spin connection formalism. The other theories we discover are shown to give examples of irregular dynamical systems. At the linear level around (anti-)de Sitter space they have two degrees of freedom, as General Relativity, with the extra ones manifesting themselves only non-linearly.
We consider the holographic duality between 4d type-A higher-spin gravity and a 3d free vector model. It is known that the Feynman diagrams for boundary correlators can be encapsulated in an HS-algebraic twistorial expression. This expression can be evaluated not just on separate boundary insertions, but on entire finite source distributions. We do so for the first time, and find that the result Z_HS disagrees with the usual CFT partition function. While such disagreement was expected due to contact corrections, it persists even in their absence. We ascribe it to a confusion between on-shell and off-shell boundary calculations. In Lorentzian boundary signature, this manifests via wrong relative signs for Feynman diagrams with different permutations of the source points. In Euclidean, the signs are instead ambiguous, spoiling would-be linear superpositions. Framing the situation as a conflict between boundary locality and HS symmetry, we sacrifice locality and choose to take Z_HS seriously. We are rewarded by the dissolution of a long-standing pathology in higher-spin dS/CFT. Though we lose the connection to the local CFT, the precise form of Z_HS can be recovered from first principles, by demanding a spin-local boundary action.
In this work we classify homogeneous solutions to the Noether procedure in (A)dS for an arbitrary number of external legs and in general dimensions. We also give a review of the corresponding flat space classification and its relation with the (A)dS result presented here. The role of dimensional dependent identities is also investigated.
We study an $SO(1,3)$ pure connection formulation in four dimensions for real-valued fields, inspired by the Capovilla, Dell and Jacobson complex self-dual approach. By considering the CMPR BF action, also, taking into account a more general class of the Cartan-Killing form for the Lie algebra $mathfrak{so(1,3)}$ and by refining the structure of the Lagrange multipliers, we integrate out the metric variables in order to obtain the pure connection action. Once we have obtained this action, we impose certain restrictions on the Lagrange multipliers, in such a way that the equations of motion led us to a family of torsionless conformally flat Einstein manifolds, parametrized by two numbers. Finally, we show that, by a suitable choice of parameters, that self-dual spaces (Anti-) De Sitter can be obtained.
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