The ͑m 2 , L͒ plane of spin s . 1 massive fields in (anti-) de Sitter space backgrounds is shown to consist of separate phases, divided by lines of novel "partially massless" gauge theories that successively remove helicities, starting from the lowest, 0 or 61͞2. The norms of the excluded states flip as the gauge lines are crossed and only the region containing the massive Minkowski theory is unitary. The partially massless gauge theories are unitary or not, depending on the ordering of the gauge lines. This "level splitting" of massless Minkowski gauge theories is specific to nonzero L. DOI: 10.1103/PhysRevLett.87.031601 PACS numbers: 11.10.Kk, 03.65.Pm, 04.62. +v, 11.15. -q The search for a Higgs as generator of mass is a central endeavor of modern particle physics. At the same time there is mounting evidence that our universe has a nonvanishing cosmological constant [1]. It is therefore extremely important to understand what mass means in cosmological backgrounds, also as string theory in cosmological backgrounds has led to significant theoretical advances. We find that conventional flat space notions must be extended for spins s . 1. Although no fundamental s . 1 particles have been observed and there is a large disparity between cosmological and nuclear scales, our results are relevant to all these research areas.Gauge invariance, masslessness, null propagation, and strictly helicity 6s excitations are all synonymous in flat space, a degeneracy that is lifted in (anti-) de Sitter [(A)dS]. We show that, for massive spin s . 1 theories, new gauge invariances allow "partially massless" propagation. Specifically, there are intermediate theories with 2, 4, . . . , 2s, 2s 1 1 propagating degrees of freedom (PDOF) for bosons or 2, 4, . . . , 2s PDOF for fermions.[The term helicity is used in a correspondence sense with Minkowski space. Also, a new feature for s $ 5͞2 is that auxiliary fields are necessary: in the massless limit they propagate but decouple. In passing, we note that conformal null propagation is generically also lost in (A)dS [2].]The physics of massive higher spin fields in (A)dS is best displayed in the ͑m 2 , L͒ plane where, phases describing 2s 1 1 massive PDOF are separated by lines where gauge invariances remove subsets of lowest helicity modes. Further, the norms of the excised helicities flip sign as these gauge lines are crossed. As a consequence, (i) only the phase occupied by the flat massive theory is unitary, since the set of excitations flipping sign is distinct on each line. (ii) Unitarity of the partially massless theories requires that their gauge invariances remove any negative norm states. This occurs only when, starting from the unitary Minkowski region, the line removing the lowest helicity state(s) can be traversed first. Subsequent lines must also be traversed in order, ending on the strictly massless helicity 6s line. Each Bianchi identity implies gauge invariances which remove corresponding lower helicities from the spectrum. In addition, the coefficients of the divergen...

Massive spin s ≥ 3/2 fields can become partially massless in cosmological backgrounds. In the plane spanned by m 2 and Λ, there are lines where new gauge invariances permit intermediate sets of higher helicities, rather than the usual flat space extremes of all 2s + 1 massive or just 2 massless helicities. These gauge lines divide the (m 2 , Λ) plane into unitarily allowed or forbidden intermediate regions where all 2s+1 massive helicities propagate but lower helicity states can have negative norms. We derive these consequences for s = 3/2, 2 by studying both their canonical (anti)commutators and the transmutation of massive constraints to partially massless Bianchi identities. For s = 2, a Hamiltonian analysis exhibits the absence of zero helicity modes in the partially massless sector. For s = 5/2, 3 we derive Bianchi identities and their accompanying gauge invariances for the various partially massless theories with propagating helicities (±5/2, ±3/2) and (±3, ±2), (±3, ±2, ±1), respectively. Of these, only the s = 3 models are unitary. To these ends, we also provide the half integer generalization of the integer spin wave operators of Lichnerowicz. Partial masslessness applies to all higher spins in (A)dS as seen by their degree of freedom counts. Finally a derivation of massive d = 4 constraints by dimensional reduction from their d = 5 massless Bianchi identity ancestors is given.

We propose a program for counting microstates of four-dimensional BPS black holes in N ≥ 2 supergravities with symmetric-space valued scalars by exploiting the symmetries of timelike reduction to three dimensions. Inspired by the equivalence between the four dimensional attractor flow and geodesic flow on the three-dimensional scalar manifold, we radially quantize stationary, spherically symmetric BPS geometries. Connections between the topological string amplitude, attractor wave function, the Ooguri-Strominger-Vafa conjecture and the theory of automorphic forms suggest that black hole degeneracies are counted by Fourier coefficients of modular forms for the threedimensional U-duality group, associated to special "unipotent" representations which appear in the supersymmetric Hilbert space of the quantum attractor flow.

We show explicitly that all partially and strictly massless fields with spins s ≤ 3 in (A)dS have null propagation. Assuming that this property holds also for s > 3, we derive the mass-cosmological constant tunings required to yield all higher spin partially massless theories. As s increases, the unitarily allowed region for massive spins is squeezed around Λ = 0, so that an infinite tower of massive particles forces vanishing Λ. We also speculate on the relevance of this result to string theory and supergravity in (A)dS backgrounds.

We analyze the physics of massive spin 2 fields in (A)dS backgrounds and exhibit that: The theory is stable only for masses m 2 ≥ 2Λ/3, where the conserved energy associated with the background timelike Killing vector is positive, while the instability for m 2 < 2Λ/3 is traceable to the helicity 0 energy. The stable, unitary, partially massless theory at m 2 = 2Λ/3 describes 4 propagating degrees of freedom, corresponding to helicities (±2, ±1) but contains no 0 helicity excitation.