Constraints on Long-Range Forces in De Sitter Space

TL;DR

The paper investigates the consistency of partially massless (PM) fields in de Sitter space by translating bulk coupling questions into boundary correlation constraints generated by partially conserved currents. Using a charge-conservation framework that combines current algebras with conformal three-point structures, the authors derive strong no-go results for PM spin-2 and spin-3 fields coupled to gravity in four dimensions, unless additional (often non-unitary) fields are introduced. In higher dimensions, they identify candidate, potentially unitary spectra that require extra PM fields, hinting at an infinite tower akin to higher-spin algebras. The work also explores Higgsed PM symmetry and double-trace deformations, demonstrating how boundary conditions can softly break PM conservation and induce anomalous dimensions, thereby delineating the landscape of consistent PM theories in cosmological spacetimes.

Abstract

The representation theory of de Sitter space admits partially massless (PM) particles, but whether such particles can participate in consistent interacting theories remains unclear. We investigate the consistency of theories containing PM fields, particularly when these fields are coupled to gravity. Our strategy exploits the fact that PM fields correspond to partially conserved currents on the spacetime boundary, which generate symmetries. These symmetries place stringent constraints on correlation functions of charged operators, allowing us to test the consistency of a proposed bulk spectrum. When the assumed operator content violates these constraints, the corresponding bulk theory is ruled out. Applying this framework, we show that, in four-dimensional de Sitter space, PM fields of spin 2 or 3 (at depth 0) cannot couple consistently to gravity: such couplings necessitate additional massive fields, which are inevitably non-unitary. In higher dimensions, however, the constraints can be satisfied without violating unitarity if further PM fields are included. The resulting structure leads to additional charge conservation laws, which suggests that consistency may ultimately require an infinite tower of higher-spin PM fields, akin to the situation for ordinary higher-spin symmetries. The methods developed here provide powerful constraints on possible long-range interactions in de Sitter space and delineate the landscape of consistent quantum field theories in cosmological spacetimes.

Figures (3)

• Figure 1: Unitary bosonic representations of dS$_{4}$. The solid dots denote the mass and spin values of the exceptional type-II series, corresponding to (partially) massless fields. The blue lines correspond to massive fields in the complementary and principal series.
• Figure 2: Insertion of a charge operator $Q[\Sigma]$ on a surface $\Sigma$ linking all local operator insertions. In the right-hand figure the surface is deformed to act locally on each operator $[Q,O(x)]$. In this image the boundary CFT is depicted as being defined on $S^d$, the conformal compactification of $\mathds{R}^d$. Deforming the surface in the left-hand figure to contract on the "other side" of the sphere, the charge operator measures the charge of the vacuum, taken to be zero by assumption.
• Figure 3: Illustration of the candidate spectra for $d=3$ ( left) and $d = 7$ ( right). In $d=3$, the masses of the fields dual to the operators $X_{(2,0)}$ and $X_{(2,-1)}$ coincide---the latter is marginally non-unitary, leaving a unique candidate spectrum with the additional operators $\{X_{(3,0)},X_{(3,1)}\}$. In $d>3$, candidate consistent spectra exist where the additional operators are either $X_{(2,-1)}$ or $\{X_{(3,0)},X_{(3,1)}\}$.