Past Relativity Seminar

We will discuss the origin of the conjectured colour-kinematics

duality in perturbative gauge theory, according to which there is a

symmetry between the colour dependence and the kinematic dependence of the

S-matrix. Based on this duality, there is a prescription to obtain gravity

amplitudes as the "double copy" of gauge theory amplitudes. We will first

consider tree-level amplitudes, where a diffeomorphism algebra underlies

the structure of MHV amplitudes, mirroring the colour algebra. We will

then draw on the progress at tree-level to consider one-loop amplitudes.

Abstract: In this talk, I will discuss the proportionality between tree amplitudes and the ultraviolet divergences in their one-loop corrections in Yang-Mills and (N < 4) Super Yang-Mills theories in four-dimensions. From the point of view of local perturbative quantum field theory, i.e. Feynman diagrams, this proportionality is straightforward: ultraviolet divergences at loop-level are absorbed into coefficients of local operators/interaction vertices in the original tree-amplitude. Ultraviolet divergences in loop amplitudes are also calculable through on-shell methods. These methods ensure manifest gauge-invariance, even at loop-level (no ghosts), at the expense of manifest locality. From an on-shell perspective, the proportionality between the ultraviolet divergences the tree amplitudes is thus not guaranteed. I describe systematic structures which ensure proportionality, and their possible connections to other recent developments in the field.

Solitons are localised non-singular lumps of energy which describe particles non perturbatively. Finding the solitons usually involves solving nonlinear differential equations, but I shall show that in some cases the solitons emerge directly from the underlying space-time geometry: certain abelian vortices arise from surfaces of constant mean curvature in Minkowski space, and skyrmions can be constructed from the holonomy of gravitational instantons.

Abstract: In this talk, I will discuss the peeling behaviour of the Weyl tensor near null infinity for asymptotically flat higher dimensional spacetimes. The result is qualitatively different from the peeling property in 4d. Also, I will discuss the rewriting of the Bondi energy flux in terms of "Newman-Penrose" Weyl components.

Abstract:

Motivated by the correlation functions-Wilson loop correspondence in maximally supersymmetric Yang-Mills theory, we will investigate a conjecture of Alday, Buchbinder, and Tseytlin regarding correlators of null polygonal Wilson loops with local operators in general position.  By translating the problem to twistor space, we can show that such correlators arise by taking null limits of correlation functions in the gauge theory, thereby providing a proof for the conjecture.  Additionally, twistor methods allow us to derive a recursive formula for computing these correlators, akin to the BCFW recursion for scattering amplitudes.

The WZW functional for a map from a surface to a Lie group has a role in the theory of harmonic maps, and it also arises as the determinant of a d-bar operator on the surface, as the action functional for a 2-dimensional quantum field theory, as the partition function of 3-dimensional Chern-Simons theory on a manifold with boundary, and as the norm-squared of a state-vector. It is intimately related to the quantization of the symplectic manifold of flat bundles on the surface, a fascinating test-case for different approaches to geometric quantization. It is also interesting as an example of interpolation between commutative and noncommutative geometry. I shall try to give an overview of the area, focussing on the aspects which are still not well understood.

I shall start with an idea (somewhat heuristic) that I call `Thermal Holography' and use that to probe the thermal behaviour of quantum horizons, i.e., without using any classical geometry, but using ordinary statistical mechanics with Gaussian fluctuations. This approach leads to a criterion for thermal stability for thermally active horizons with an Isolated horizon as an equilibrium configuration, whose (microcanonical) entropy has been computed using Loop Quantum Gravity (I shall outline this computation). As fiducial checks, we briefly look at some very well-known classical black hole metrics for their thermal stability and recover known results. Finally, I shall speculate about a possible link between our stability criterion and the Chandrasekhar upper bound for the mass of stable neutron stars.

This talk will be based on arxiv:1106.5254.

This talk will be based on arxiv:1106.5254.

There have been significant progress in the calculation of scattering amplitudes in N=4 SYM. In this talk I will consider `form factors', which are defined not only with on-shell asymptotic states but also with one off-shell operator inserted. Such quantities are in some sense the hybrid of on-shell quantities (such

as scattering amplitudes) and off-shell quantities (such as correlation functions). We will see that form factors inherit many nice properties of scattering amplitudes, in particular we will consider their supersymmetrization and the dual picture.

In field theory simple forms of certain scattering amplitudes in four dimensional theories with massless particles are known. This has been shown to be closely related to underlying (super)symmetries and has been a source of inspiration for much development in the last years. Away from four dimensions much less is known with some concrete development only in six dimensions. I will show how to construct promising on-shell superspaces in eight and ten dimensions which permit suggestively simple forms of supersymmetric four point scattering amplitudes with massless particles. Supersymmetric on-shell recursion relations which allow one to compute in principle any amplitude are constructed, as well as the three point `seed' amplitudes to make these work. In the three point case I will also present some classes of supersymmetric amplitudes with a massive particle for the type IIB superstring in a flat background.

Consider the space M of parabolas y=ax^2+bx+c, with (a, b, c) as coordinates on M. Two parabolas generically intersect at two (possibly complex) points, and we can define a conformal structure on M by declaring two points to be null separated iff the corresponding parabolas are tangent. A simple calculation of discriminant shows that this conformal structure is flat.

In this talk (based on joint works with Godlinski and Sokolov) I shall show how replacing parabolas by rational plane curves of higher degree allows constructing curved conformal structures in any odd dimension. In dimension seven one can use this "twistor" construction to find G_2 structures in a conformal class.

A celebrated result in mathematical general relativity is the uniqueness of the Kerr(-Newman) black-holes as regular solutions to the stationary and axially-symmetric Einstein(-Maxwell) equations. The axial symmetry can be removed if one invokes Hawking's rigidity theorem. Hawking's theorem requires, however, real analyticity of the solution. A recent program of A. Ionescu and S. Klainerman seeks to remove the analyticity requirement in the vacuum case. They were able to show that any smooth extension of "Kerr data" prescribed on the horizon, satisfying the Einstein vacuum equations, must be Kerr, using a characterization of Kerr metric due to M. Mars. In this talk I will give a characterization for the Kerr-Newman metric, and extend the rigidity result to cover the electrovacuum case.

I will outline two areas currently under study by myself and my co-workers, particularly Jonathan Holland: one concerns the relation between the exceptional Lie group G_2 and Einstein's gravity; the second will introduce and apply the concept of a causal geometry.

We consider Einstein-scalar field Lichnerowicz equations in the positive case in compact Riemannian manifolds. We discuss existence and stability issues for these equations

Canonical quantization of gravitational field will beconsidered. Examples for which the procedure can be completed (without reducingthe degrees of freedom) will be presented and discussed. The frameworks appliedwill be: Loop Quantum Gravity, relational construction of the Dirac observablesand deparametrization.

I will present existence and uniqueness results for theCauchy problem as in the title.

Extremal black holes are of interest as they are expected have simpler quantum descriptions than their non-extremal counterparts.  Any extremal black hole solution admits a well defined notion of a near horizon geometry which solves the same field equations. I will describe recent progress on the general understanding of such near horizon geometries in four and higher dimensions. This will include the proof of near-horizon symmetry enhancement and the explicit classification of near-horizon geometries (in a variety of settings). I will also discuss how one can use such results to prove classification/uniqueness theorems for asymptotically flat extremal vacuum black holes in four and five dimensions.

Algebraic classification of the Weyl tensor is an important tool for solving the Einstein equation. I shall review the classification for spacetimes of dimension greater than four, and recent progress in using it to construct new exact solutions. The higher-dimensional generalization of the Goldberg-Sachs theorem will be discussed.

Two classes of solutions of the Einstein equations with symmetry which

are frequently studied are the Bianchi and Gowdy models. The aim of this

talk is to explain certain relations between these two classes of

spacetimes which can provide insights into the dynamics of both. In

particular it is explained that the special case of the Gowdy models known as circular loop spacetimes are Bianchi models in disguise. Generalizations of Gowdy spacetimes which can be thought of as inhomogeneous perturbations of some of the Bianchi models are introduced.

Results concerning their dynamics are presented.