Vielbein Formalism Research Articles

New kinetic terms for massive gravity and multi-gravity: a no-go in vielbein form

We reconsider the possibility of a class of new kinetic terms in the first order (vielbein) formulation of massive gravity and multi-gravity. We find that new degrees of freedom emerge which are not associated with the Boulware–Deser ghost and are intrinsic to the vielbein formulation. These new degrees of freedom are associated with the Lorentz transformations which encode the additional variables contained in the vielbein over the metric. Although they are not guaranteed to be ghostly, they are nevertheless infinitely strongly coupled on Minkowski spacetime and are not part of the spin-2 multiplet. Hence their existence implies the uniqueness of the Einstein–Hilbert term as the kinetic term for a massive graviton.

Super Virasoro Algebras From Chiral Supergravity

In this note, we construct Noether charges for the chiral supergravity, which contains the Lorentz Chern–Simons term, by applying Wald’s prescription to the vielbein formalism. We investigate the AdS3/CFT2 correspondence by using the vielbein formalism. The asymptotic symmetry group is carefully examined by taking into account the local Lorentz transformation, and we construct super Virasoro algebras with central extensions from the chiral supergravity.

Vielbein to the rescue? Breaking the symmetric vielbein condition in massive gravity and multigravity

Non-minimal matter couplings have recently been considered in the context of massive gravity and multi-gravity. These couplings are free of the Boulware-Deser ghost in the decoupling limit and can thus be considered within an Effective Field Theory setup. Beyond the decoupling limit the ghost was shown to reemerge in the metric formulation of the theory. Recently it was argued that this pathology is absent when formulated in terms of unconstrained vielbeins. We investigate this possibility and show that the Boulware-Deser ghost is always present beyond the decoupling limit in any dimension larger than two. We also show that the metric and vielbein formulations have an identical ghost-free decoupling limit. Finally we extend these arguments to more generic multi-gravity theories and argue that for any dimension larger than two a ghost is also present in the vielbein formulation whenever the symmetric vielbein condition is spoiled and the equivalence with the metric formulation is lost.

Spin-base invariance of fermions in arbitrary dimensions

The concept of spin-base invariance is extended to arbitrary integer dimension $d \geq 2$. Explicit formulas for the spin connection as a function of the Dirac matrices are found. We disclose the hidden spin-base invariance of the vielbein formalism and give a detailed motivation for this symmetry from first principles. The common Lorentz symmetric gauge for the vielbein is constructed for the Dirac matrices, even for metrics which are not linearly connected. Under certain criteria, it constitutes the simplest possible gauge, demonstrating why this gauge is so useful.

Global surpluses of spin-base invariant fermions

The spin-base invariant formalism of Dirac fermions in curved space maintains the essential symmetries of general covariance as well as similarity transformations of the Clifford algebra. We emphasize the advantages of the spin-base invariant formalism both from a conceptual as well as from a practical viewpoint. This suggests that local spin-base invariance should be added to the list of (effective) properties of (quantum) gravity theories. We find support for this viewpoint by the explicit construction of a global realization of the Clifford algebra on a 2-sphere which is impossible in the spin-base non-invariant vielbein formalism.

The coupling to matter in massive, bi- and multi-gravity

In this paper we construct a family of ways in which matter can couple to one or more `metrics'/spin-2 fields in the vielbein formulation. We do so subject to requiring the weak equivalence principle and the absence of ghosts from pure spin-2 interactions generated by the matter action. Results are presented for Massive, Bi- and Multi-Gravity theories and we give explicit expressions for the effective matter metric in all of these cases.

General coordinate invariance in quantum many-body systems

We extend the notion of general coordinate invariance to many-body, not necessarily relativistic, systems. As an application, we investigate nonrelativistic general covariance in Galilei-invariant systems. The peculiar transformation rules for the background metric and gauge fields, first introduced by Son and Wingate in 2005 and refined in subsequent works, follow naturally from our framework. Our approach makes it clear that Galilei or Poincare symmetry is by no means a necessary prerequisite for making the theory invariant under coordinate diffeomorphisms. General covariance merely expresses the freedom to choose spacetime coordinates at will, whereas the true, physical symmetries of the system can be separately implemented as "internal" symmetries within the vielbein formalism. A systematic way to implement such symmetries is provided by the coset construction. We illustrate this point by applying our formalism to nonrelativistic s-wave superfluids.

Hamiltonian and Noether charges in first order gravity

We consider gravity in four dimensions in the vielbein formulation, where the fundamental variables are a tetrad $e$ and a SO(3,1) connection $\omega$. We start with the most general action principle compatible with diffeomorphism invariance which includes, besides the standard Palatini term, other terms that either do not change the equations of motion, or are topological in nature. For our analysis we employ the covariant Hamiltonian formalism where the phase space $\Gamma$ is given by solutions to the equations of motion. We consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. For this extended action we study the effect of the topological terms on the Hamiltonian formulation. We prove two results. The first one is rather generic, applicable to any field theory with boundaries: The addition of topological terms (and any other boundary term) does not modify the symplectic structure of the theory. The second result pertains to the conserved Hamiltonian and Noether charges, whose properties we analyze in detail, including their relationship. While the Hamiltonian charges are unaffected by the addition of topological and boundary terms, we show in detail that the Noether charges {\em do} change. Thus, a non-trivial relation between these two sets of charges arises when the boundary and topological terms needed for a consistent formulation are included.

Torsional Newton-Cartan geometry and Lifshitz holography

We obtain the Lifshitz UV completion in a specific model for z=2 Lifshitz geometries. We use a vielbein formalism which enables identification of all the sources as leading components of well-chosen bulk fields. We show that the geometry induced from the bulk onto the boundary is a novel extension of Newton-Cartan geometry with a specific torsion tensor. We explicitly compute all the vevs including the boundary stress-energy tensor and their Ward identities. After using local symmetries/Ward identities the system exhibits 6+6 sources and vevs. The FG expansion exhibits, however, an additional free function which is related to an irrelevant operator whose source has been turned off. We show that this is related to a second UV completion.

Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography

For a specific action supporting z=2 Lifshitz geometries we identify the Lifshitz UV completion by solving for the most general solution near the Lifshitz boundary. We identify all the sources as leading components of bulk fields which requires a vielbein formalism. This includes two linear combinations of the bulk gauge field and timelike vielbein where one asymptotes to the boundary timelike vielbein and the other to the boundary gauge field. The geometry induced from the bulk onto the boundary is a novel extension of Newton-Cartan geometry that we call torsional Newton-Cartan (TNC) geometry. There is a constraint on the sources but its pairing with a Ward identity allows one to reduce the variation of the on-shell action to unconstrained sources. We compute all the vevs along with their Ward identities and derive conditions for the boundary theory to admit conserved currents obtained by contracting the boundary stress-energy tensor with a TNC analogue of a conformal Killing vector. We also obtain the anisotropic Weyl anomaly that takes the form of a Horava-Lifshitz action defined on a TNC geometry. The Fefferman-Graham expansion contains a free function that does not appear in the variation of the on-shell action. We show that this is related to an irrelevant deformation that selects between two different UV completions.

Translation invariant time-dependent solutions to massive gravity

Homogeneous time-dependent solutions of massive gravity generalise the plane wave solutions of the linearised Fierz-Pauli equations for a massive spin-two particle, as well as the Kasner solutions of General Relativity. We show that they also allow a clear counting of the degrees of freedom and represent a simplified framework to work out the constraints, the equations of motion and the initial value formulation. We work in the vielbein formulation of massive gravity, find the phase space resulting from the constraints and show that several disconnected sectors of solutions exist some of which are unstable. The initial values determine the sector to which a solution belongs. Classically, the theory is not pathological but quantum mechanically the theory may suffer from instabilities. The latter are not due to an extra ghost-like degree of freedom.

DIRAC EQUATION IN NON-RIEMANNIAN GEOMETRIES

We present the Dirac equation in a geometry with torsion and non-metricity balancing generality and simplicity as much as possible. In doing so, we use the vielbein formalism and the Clifford algebra. We also use an index-free formalism which allows us to construct objects that are totally invariant. It turns out that the previous apparatuses not only make possible a simple deduction of the Dirac equation but also allow us to exhibit some details that is generally obscure in the literature.

Interacting spin-2 fields

We construct consistent theories of multiple interacting spin-2 fields in arbitrary spacetime dimensions using a vielbein formulation. We show that these theories have the additional primary constraints needed to eliminate potential ghosts, to all orders in the fields, and to all orders beyond any decoupling limit. We postulate that the number of spin-2 fields interacting at a single vertex is limited by the number of spacetime dimensions. We then show that, for the case of two spin-2 fields, the vielbein theory is equivalent to the recently proposed theories of ghost-free massive gravity and bi-metric gravity. The vielbein formulation greatly simplifies the proof that these theories have an extra primary constraint which eliminates the Boulware-Deser ghost.

Stringy differential geometry, beyond Riemann

While the fundamental object in Riemannian geometry is a metric, closed string theories call for us to put a two-form gauge field and a scalar dilaton on an equal footing with the metric. Here we propose a novel differential geometry which treats the three objects in a unified manner, manifests not only diffeomorphism and one-form gauge symmetry but also O(D,D) T-duality, and enables us to rewrite the known low energy effective action of them as a single term. Further, we develop a corresponding vielbein formalism and gauge the internal symmetry which is given by a direct product of two local Lorentz groups, SO(1,D-1) times SO(1,D-1). We comment that the notion of cosmological constant naturally changes.

Half-Differentials versus Spinor Formalism for Fermions in Low-Dimensional Systems

The description of fermions on curved manifolds or in curvilinear coordinates usually requires a vielbein formalism to define Dirac γ-matrices or Pauli matrices on the manifold. Derivatives of the vielbein also enter equations of motion for fermions through the spin connection, which gauges local rotations or Lorentz transformations of tangent planes.

A vielbein formulation of unified Einstein–Maxwell theory

In the framework of -bundles a vielbein formulation of unified Einstein–Maxwell theory is proposed. In the resulting scheme, field equations matching the gravitational and electromagnetic fields are derived by constraining a five-dimensional variational principle. No dynamical scalar field is involved.

Electrons in curved low-dimensional systems: spinors or half-order differentials?

The description of fermions on curved manifolds or in curvilinear coordinates usually requires a vielbein formalism to define Dirac γ-matrices or Pauli matrices on the manifold. Derivatives of the vielbein also enter equations of motion for fermions through the spin connection, which gauges local rotations or Lorentz transformations of tangent planes. The present paper serves a dual purpose. First we will see how the zweibein formalism on surfaces emerges from constraining fermions to submanifolds of Minkowski space. However, it is known e.g. in superstring theory, that so called half-order differentials can also be used to describe fermions in two dimensions. Therefore, in the second part, I will explain how in two dimensions the zweibein can be absorbed into the spinors to form half-order differentials. The interesting point about half-order differentials is that their derivative terms along a two-dimensional submanifold of Minkoski space look exactly like ordinary spinor derivatives in Cartesian coordinates on a planar surface, and the whole effect of the background geometry reduces to a universal factor multiplying orthogonal derivative terms and mass terms.

BIANCHI IX GROUP-MANIFOLD REDUCTIONS OF GRAVITY

We exhibit a new way to perform the group-manifold reduction of pure Einstein gravity in the vielbein formulation when the compactification group manifold is S3. The new Bianchi IX group-manifold reduction is obtained by exploiting the two three-dimensional Lie algebras that the S3 group manifold admits. As an application of the new reduction we show that there exists a domain wall solution to the lower-dimensional theory which upon uplifting to the higher-dimension turns out to be the self-dual (in the nonvanishing components of both curvature and spin connection) Kaluza–Klein monopole.

S3 Group-manifold reduction of gravity

In this contribution we exhibit a new consistent group-manifold reduction of pure Einstein gravity in the vielbein formulation when the compactification group manifold is S3. The novel feature in the reduction is to consider the two 3-dimensional Lie algebras that S3 admits. We discuss the characteristics of the lower-dimensional theory and we emphasize the results generated by the new group-manifold reduction. As an application we show that the lower-dimensional theory admits a domain wall solution which upon uplifting to the higherdimension results to be the self-dual (in both curvature and spin connection) Kaluza-Klein monopole.

Symmetries and observables in topological gravity

After a brief review of topological gravity, we present a superspace approach to this theory. This formulation allows us to recover in a natural manner various known results and to gain some insight into the precise relationship between different approaches to topological gravity. Though the main focus of our work is on the vielbein formalism, we also discuss the metric approach and its relationship with the former formalism.