Aether, dark energy and string compactifications.

The nineteenth-century aether died with special relativity but was resurrected by general relativity in the form of dark energy; a tensile material with tension equal to its energy density. Such a material is provided by the D-branes of string-theory; these can support the fields of supersymmetric particle-physics, although their energy density is cancelled by orientifold singularities upon compactification. Dark energy can still arise from supersymmetry-breaking anti-D-branes but it is probably time-dependent. Recent results on time-dependent compactifications to an FLRW universe with late-time accelerated expansion are reviewed. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.

An interacting conformal chiral 2-form electrodynamics in six dimensions.

The strong-field limit for the 2-form potential on an M5-brane yields a conformal chiral 2-form electrodynamics in six dimensions, with gauge-invariant self-interactions but no adjustable coupling constant; the stress tensor is that of a null fluid. Lorentz invariance can be made manifest via an interpretation as a tensionless 'space-filling M5-brane', or as a truncation of the infrared dynamics of an M5-brane in AdS7 × S 4.

Manifestly Lorentz Invariant Chiral Boson Action.

A manifestly Lorentz invariant action is found for the Floreanini-Jackiw chiral boson. The method involves a novel chiral reduction of the phase-space action for a string and can be adapted to describe chiral bosons on the heterotic string worldsheet. A similar manifestly Lorentz invariant action is found for an entire class of conformal chiral 2k-form electrodynamics in (4k+2) dimensions which includes the Floreanini-Jackiw theory as the k=0 case.

Gravity and the Spin-2 Planar Schrödinger Equation.

A Schrödinger equation proposed for the Girvin-MacDonald-Platzman gapped spin-2 mode of fractional quantum Hall states is found from a novel nonrelativistic limit, applicable only in 2+1 dimensions, of the massive spin-2 Fierz-Pauli field equations. It is also found from a novel null reduction of the linearized Einstein field equations in 3+1 dimensions, and in this context a uniform distribution of spin-2 particles implies, via a Brinkmann-wave solution of the nonlinear Einstein equations, a confining harmonic oscillator potential for the individual particles.

Anti-de Sitter Particles and Manifest (Super)Isometries.

Starting from the classical action for a spin-zero particle in a D-dimensional anti-de Sitter (AdS) spacetime, we recover the Breitenlohner-Freedman bound by quantization. For D=4, 5, 7 and using an Sl(2;K) spinor notation for K=R,C,H, we find a bitwistor form of the action for which the AdS isometry group is linearly realized, although only for zero mass when D=4, 7 in agreement with previous constructions. For zero mass and D=4 the conformal isometry group is linearly realized. We extend these results to the superparticle in the maximally supersymmetric "AdS×S" string or M-theory vacua, showing that quantization yields a 128+128 component supermultiplet. We also extend them to the null string.

Yang-Mills as massive Chern-Simons theory: a third way to three-dimensional gauge theories.

The Yang-Mills (YM) equation in three spacetime dimensions (3D) can be modified to include a novel parity-preserving interaction term, with an inverse mass parameter, in addition to a possible topological mass term. The novelty is that the modified YM equation is not the Euler-Lagrange equation of any gauge-invariant local action for the YM gauge potential alone. Instead, consistency is achieved in the "third way" exploited by 3D minimal massive gravity. We relate our results to the "novel Higgs mechanism" for Chern-Simons gauge theories.

Zwei-dreibein gravity: a two-frame-field model of 3D massive gravity.

We present a generally covariant and parity-invariant two-frame field ("zwei-dreibein") action for gravity in three space-time dimensions that propagates two massive spin-2 modes, unitarily, and we use Hamiltonian methods to confirm the absence of unphysical degrees of freedom. We show how zwei-dreibein gravity unifies previous "3D massive gravity" models and extends them, in the context of the AdS/CFT correspondence, to allow for a positive central charge consistent with bulk unitarity.

Thermodynamics of "exotic" Bañados-Teitelboim-Zanelli black holes.

A number of three-dimensional (3D) gravity models, such as 3D conformal gravity, admit "exotic" black hole solutions: the metric is the same as the Bañados-Teitelboim-Zanelli metric of 3D Einstein gravity but with reversed roles for mass and angular momentum, and an entropy proportional to the length of the inner horizon instead of the event horizon. Here we show that the Bañados-Teitelboim-Zanelli solutions of the exotic 3D Einstein gravity (with parity-odd action but Einstein field equations) are exotic black holes, and we investigate their thermodynamics. The first and second laws of black hole thermodynamics still apply, and the entropy still has a statistical interpretation.

Anyons from strings.

The Nambu-Goto string in a three-dimensional (3D) Minkowski spacetime is quantized preserving Lorentz invariance and parity. The spectrum of massive states contains anyons. An ambiguity in the ground state energy is resolved by the 3D N=1 Green-Schwarz superstring, which has massless ground states describing a dilaton and dilatino, and first-excited states of spin 1/4.

Massive gravity in three dimensions.

A particular higher-derivative extension of the Einstein-Hilbert action in three spacetime dimensions is shown to be equivalent at the linearized level to the (unitary) Pauli-Fierz action for a massive spin-2 field. A more general model, which also includes "topologically-massive" gravity as a special case, propagates the two spin-2 helicity states with different masses. We discuss the extension to massive N-extended supergravity, and we present a "cosmological" extension that admits an anti-de Sitter vacuum.

Landau levels and Riemann zeros.

The number N(E) of complex zeros of the Riemann zeta function with positive imaginary part less than E is the sum of a "smooth" function N[over ](E) and a "fluctuation." Berry and Keating have shown that the asymptotic expansion of N[over ](E) counts states of positive energy less than E in a "regularized" semiclassical model with classical Hamiltonian H=xp. For a different regularization, Connes has shown that it counts states "missing" from a continuum. Here we show how the "absorption spectrum" model of Connes emerges as the lowest Landau level limit of a specific quantum-mechanical model for a charged particle on a planar surface in an electric potential and uniform magnetic field. We suggest a role for the higher Landau levels in the fluctuation part of N(E).

Hidden supersymmetry of domain walls and cosmologies.

We show that all domain-wall solutions of gravity coupled to scalar fields for which the world-volume geometry is Minkowski or anti-de Sitter admit Killing spinors, and satisfy corresponding first-order equations involving a superpotential determined by the solution. By analytic continuation, all flat or closed Friedmann-Lemaître-Robertson-Walker cosmologies are shown to satisfy similar first-order equations arising from the existence of "pseudo Killing" spinors.

Open M5-branes.

We show how, in heterotic M theory, an M5-brane in the 11-dimensional bulk may end on an "M9-brane" boundary, the M5-brane boundary being a Yang-monopole 4-brane. This possibility suggests various novel 5-brane configurations of heterotic M theory, in particular, a static M5-brane suspended between the two M9-brane boundaries, for which we find the asymptotic heterotic supergravity solution.

Accelerating cosmologies from compactification.

A solution of the (4+n)-dimensional vacuum Einstein equations is found for which spacetime is compactified on an n-dimensional compact hyperbolic manifold (n> or =2) of time-varying volume to a flat four-dimensional Friedmann-Lemaitre-Robertson-Walker cosmology undergoing a period of accelerated expansion in the Einstein conformal frame. This shows that the "no-go" theorem forbidding acceleration in "standard" (time-independent) compactifications of string or M theory does not apply to "cosmological" (time-dependent) hyperbolic compactifications.