Andrew Waldron

Partial masslessness of higher spins in (A)dS

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 (m2,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.

Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations

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.

Stability of massive cosmological gravitons

We analyze the physics of massive spin-2 fields in (A)dS backgrounds and exhibit that: The theory is stable only for masses m2⩾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.

BPS black holes, quantum attractor flows and automorphic forms

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 three-dimensional U-duality group, associated to special unipotent representations which appear in the supersymmetric Hilbert space of the quantum attractor flow.

Acausality of Massive Gravity

We show, by analyzing its characteristics, that the ghost-free, 5 degree of freedom, Wess-Zumino massive gravity model admits superluminal shock wave solutions and thus is acausal. Ironically, this pathology arises from the very constraint that removes the (sixth) Boulware-Deser ghost mode.

On Euclidean spinors and Wick rotations

Abstract We propose a continuous Wick rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space which treats fermions on the same footing as bosons. The result is a recipe to construct a supersymmetric Euclidean theory from any supersymmetric Minkowski theory. This Wick rotation is identified as a complex Lorentz boost in a five-dimensional space and acts uniformly on bosons and fermions. For Majorana and Weyl spinors our approach is reminiscent of the traditional Osterwalder-Schrader approach in which spinors are “doubled” but the action is not hermitian. However, for Dirac spinors our work provides a link to the work of Schwinger and Zumino in which hermiticity is maintained but spinors are not doubled. Our work differs from recent work by Mehta since we introduce no external metric and transform only the basic fields.

Quantum attractor flows

Motivated by the interpretation of the Ooguri-Strominger-Vafa conjecture as a holographic correspondence in the mini-superspace approximation, we study the radial quantization of stationary, spherically symmetric black holes in four dimensions. A key ingredient is the classical equivalence between the radial evolution equation and geodesic motion of a fiducial particle on the moduli space M*3 of the three-dimensional theory after reduction along the time direction. In the case of N = 2 supergravity, M*3 is a para-quaternionic-Kahler manifold; in this case, we show that BPS black holes correspond to a particular class of geodesics which lift holomorphically to the twistor space Z of M*3, and identify Z as the BPS phase space. We give a natural quantization of the BPS phase space in terms of the sheaf cohomology of Z, and compute the exact wave function of a BPS black hole with fixed electric and magnetic charges in this framework. We comment on the relation to the topological string amplitude, extensions to N > 2 supergravity theories, and applications to automorphic black hole partition functions.

Constant curvature algebras and higher spin action generating functions

The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R) [semidirect product] R^2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for the actions and gauge invariances of massive, partially massless and massless (for both bose and fermi statistics) higher spins on constant curvature backgrounds. These are formulated in terms of a minimal set of covariant, unconstrained, fields rather than towers of auxiliary fields. Partially massless gauge transformations are shown to arise as degeneracies of the flat, massless gauge transformation in one dimension higher. Moreover, our results and calculus offer a considerable simplification over existing techniques for handling higher spins. In particular, we show how theories of arbitrary spin in dimension d can be rewritten in terms of a single scalar field in dimension 2d where the d additional dimensions correspond to coordinate differentials. We also develop an analogous framework for spinor-tensor fields in terms of the corresponding superalgebra.

Minimal Representations, Spherical Vectors¶and Exceptional Theta Series

Abstract: Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrödinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the rôle of the summand in the automorphic theta series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the theta series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.

Inconsistencies of massive charged gravitating higher spins

We examine the causality and degree-of-freedom (DoF) problems encountered by charged, gravitating, massive higher spin fields. For spin s = 3/2, making the metric dynamical yields improved causality bounds. These involve only the mass, the product eMP of the charge and Planck mass and the cosmological constant Λ. The bounds are themselves related to a gauge invariance of the timelike component of the field equation at the onset of acausality. While propagation is causal in arbitrary E/M backgrounds, the allowed mass ranges of parameters are of Planck order. Generically, interacting spins s > 3/2 are subject to DoF violations as well as to acausality; the former must be overcome before analysis of the latter can even begin. Here we review both difficulties for charged s = 2 and show that while a g-factor of 1/2 solves the DoF problem, acausality persists for any g. Separately we establish that no s = 2 theory — DoF preserving or otherwise — can be tree unitary.