arXiv subjects

Yannick Herfray

Publications and source records attributed to Yannick Herfray.

27 records · Page 2Linked to original sources

arXiv2018

New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's Three-Forms

In this thesis we take Einstein theory in dimension four seriously, and explore the special aspects of gravity in this number of dimension. Among the many surprising features in dimension four, one of them is the possibility of `Chiral formulations of gravity' - they are surprising as they typically do not rely on a metric. Another is the existence of the Twistor correspondence. The Chiral and Twistor formulations might seems different in nature. In the first part of this thesis we demonstrate that they are in fact closely related. In particular we give a new proof for Penrose's `non-linear graviton theorem' that relies on the geometry of SU(2)-connections only (rather than on metric). In the second part of this thesis we describe partial results towards encoding the full GR in the total space of some fibre bundle over space-time. We indeed show that gravity theory in three and four dimensions can be related to theories of a completely different nature in six and seven dimension respectively. This theories, first advertised by Hitchin, are diffeomorphism invariant theories of differential three-forms. Starting with seven dimensions, we are only partially succesfull: the resulting theory is some deformed version of gravity. We however found that solutions to a particular gravity theory in four dimension have a seven dimensional interpretation as G2 holonomy manifold. On the other hand by going from six to three dimension we do recover three dimensional gravity. As a bonus, we describe new diffeomorphism invariant functionnals for differential forms in six dimension and prove that two of them are topological.

gr-qc↗

arXiv2018

Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity

This paper establishes the relation between traditional results from (euclidean) twistor theory and chiral formulations of General Relativity (GR), especially the pure connection formulation. Starting from a $SU(2)$-connection only we show how to construct natural complex data on twistor space, mainly an almost Hermitian structure and a connection on some complex line bundle. Only when this almost Hermitian structure is integrable is the connection related to an anti-self-dual-Einstein metric and makes contact with the usual results. This leads to a new proof of the non-linear-graviton theorem. Finally we discuss what new strategies this "connection approach" to twistors suggests for constructing a twistor action for gravity. In appendix we also review all known chiral Lagrangians for GR.

hep-th↗

arXiv2017

The complex null string, Galilean conformal algebra and scattering equations

The scattering equation formalism for scattering amplitudes, and its stringy incarnation, the ambitwistor string, remains a mysterious construction. In this paper, we pursue the study a gauged-unfixed version of the ambitwistor string known as the null string. We explore the following three aspects in detail; its complexification, gauge fixing, and amplitudes. We first study the complexification of the string; the associated symmetries and moduli, and connection to the ambitwistor string. We then look in more details at the leftover symmetry algebra of the string, called Galilean conformal algebra; we study its local and global action and gauge-fixing. We finish by presenting an operator formalism, that we use to compute tree-level scattering amplitudes based on the scattering equations and a one-loop partition function. These results hopefully will open the way to understand conceptual questions related to the loop expansion in these twistor-like string models.

hep-th↗

arXiv2017

Topological field theories of 2- and 3-forms in six dimensions

We consider several diffeomorphism invariant field theories of 2- and 3-forms in six dimensions. They all share the same kinetic term $BdC$, but differ in the potential term that is added. The theory $BdC$ with no potential term is topological - it describes no propagating degrees of freedom. We show that the theory continues to remain topological when either the $BBB$ or $C\hat{C}$ potential term is added. The latter theory can be viewed as a background independent version of the 6-dimensional Hitchin theory, for its critical points are complex or para-complex 6-manifolds, but unlike in Hitchin's construction, one does not need to choose of a background cohomology class to define the theory. We also show that the dimensional reduction of the $C\hat{C}$ theory to three dimensions, when reducing on S3, gives 3D gravity.

hep-th↗

arXiv2017

6D Interpretation of 3D Gravity

We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets an explanation of why the pure connection formulation of 3D gravity exists. More generally, we interpret 3D gravity as the dimensional reduction of the 6D Hitchin theory. To this end, we show that any SU(2) invariant closed 3-form in the total space of the principal SU(2) bundle can be parametrised by a connection together with a 2-form field on the base. The dimensional reduction of the 6D Hitchin theory then gives rise to 3D gravity coupled to a topological 2-form field.

hep-th↗

arXiv2016

Anisotropic singularities in chiral modified gravity

In four space-time dimensions, there exists a special infinite-parameter family of chiral modified gravity theories. All these theories describe just two propagating polarizations of the graviton. General Relativity with an arbitrary cosmological constant is the only parity-invariant member of this family. We review how these modified gravity theories arise within the framework of pure-connection formulation. We introduce a new convenient parametrisation of this family of theories by using certain set of auxiliary fields. Modifications of General Relativity can be arranged so as to become important in regions with large Weyl curvature, while the behaviour is indistinguishable from GR where Weyl curvature is small. We show how the Kasner singularity of General Relativity is resolved in a particular class of modified gravity theories of this type, leading to solutions in which the fundamental connection field is regular all through the space-time. There arises a new asymptotically De Sitter region `behind' the would-be singularity, the complete solution thus being of a bounce type.

gr-qc↗

arXiv2016

A 4D gravity theory and G2-holonomy manifolds

Bryant and Salamon gave a construction of metrics of G2 holonomy on the total space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional self-dual Einstein manifold. We generalise it by considering the total space of an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection on this bundle. We make essentially the same ansatz for the calibrating 3-form, but use the curvature 2-forms instead of the ASD ones. We show that the resulting 3-form defines a metric of G2 holonomy if the connection satisfies a certain second-order PDE. This is exactly the same PDE that arises as the field equation of a certain 4-dimensional gravity theory formulated as a diffeomorphism-invariant theory of SO(3) connections. Thus, every solution of this 4-dimensional gravity theory can be lifted to a G2-holonomy metric. Unlike all previously known constructions, the theory that we lift to 7 dimensions is not topological. Thus, our construction should give rise to many new metrics of G2 holonomy. We describe several examples that are of cohomogeneity one on the base.

hep-th↗

arXiv2015

Asymptotically hyperbolic connections

General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solve the arising "evolution" equations order by order in the expansion in powers of the radial coordinate. The solution in the connection setting is arguably simpler, and very straightforward algebraic manipulations allow one to see how the obstruction appears at third order in the expansion. Another interesting feature of the connection formulation is that the "counter terms" required in the computation of the renormalised volume all combine into the Chern-Simons functional of the restriction of the connection to the boundary. As the Chern-Simons invariant is only defined modulo large gauge transformations, the requirement that the path integral over asymptotically hyperbolic connections is well-defined requires the cosmological constant to be quantised. Finally, in the connection setting one can deform the 4D Einstein condition in an interesting way, and we show that asymptotically hyperbolic connection expansion is universal and valid for any of the deformed theories.

gr-qc↗

arXiv2015

New first order Lagrangian for General Relativity

We describe a new BF-type first-order in derivatives Lagrangian for General Relativity. The Lagrangian depends on a connection field as well as a Lie-algebra valued two-form field, with no other fields present. There are two free parameters, which translate into the cosmological constant and the coefficient in front of a topological term. When one of the parameters is set to zero, the theory becomes topological. When the other parameter is zero, the theory reduces to the (anti-) self-dual gravity. Thus, our new Lagrangian interpolates between the topological and anti-self-dual gravities. It also interprets GR as the (anti-) self-dual gravity with an extra quadratic in the auxiliary two-form field term added to the Lagrangian, precisely paralleling the situation in Yang-Mills theory.

gr-qc↗