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Timetable (GTAW01)

General relativity: from geometry to amplitudes

Monday 27th June 2016 to Friday 1st July 2016

Monday 27th June 2016
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from John Toland (INI Director) INI 1
10:00 to 11:00 Bernd Ammann (Universität Regensburg)
Ricci-flat manifolds and a spinorial flow
Joint work with Klaus Kröncke, Hartmut Weiß and Frederik Witt

We study the set of all Ricci-flat Riemannian metrics on a given compact manifold M.
We say that a Ricci-flat metric on M is structured if its pullback to the universal cover admits a parallel spinor. The holonomy of these metrics is special as these manifolds carry some additional structure, e.g. a Calabi-Yau structure or a G2-structure.

The set of unstructured Ricci-flat metrics is poorly understood.  Nobody knows whether unstructured compact Ricci-flat Riemannian manifolds exist, and if they exist, there is no reason to expect that the set of such metrics on a fixed compact manifold should have the structure of a smooth manifold.

On the other hand, the set of structured Ricci-flat metrics on compact manifolds is now well-understood.

The set of structured Ricci-flat metrics is an open and closed subset in the space of all Ricci-flat metrics.
The holonomy group is constant along connected components.
The dimension of the space of parallel spinors as well.
The structured Ricci-flat metrics form a smooth Banach submanifold in the space of all metrics.
Furthermore the associated premoduli space is a finite-dimensional smooth manifold.

These results build on previous work by J. Nordström, Goto, Koiso, Tian & Todorov, Joyce, McKenzie Wang and many others.
The important step is to pass from irreducible to reducible holonomy groups.

In the last part of the talk we summarize work on the L2-gradient flow of the functional $(g,\phi)\mapsto E(g,\phi):=\int_M|\nabla^g\phi|^2$.
This is a weakly parabolic flow on the space of metrics and spinors of constant unit length. The flow is supposed to flow against structured Ricci-flat metics. Its geometric interpretation in dimension 2 is some kind of Willmore flow, and in dimension 3 it is a frame flow.
We find that the functional E is a Morse-Bott functional. This fact is related to stability questions.

Associated publications:  
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Yong Wei (University College London)
Laplacian flow for closed G_2-structures
We will discuss the Laplacian flow for closed G_2 structures. This flow was introduced by R. Bryant in 1992 to study the geometry of G_2 structures, inspired by Hamilton's Ricci flow in studying the generic Riemannian structures and the Kahler Ricci flow in studying Kahler structures. The primary goal is to understand the conditions under which the Laplacian flow can converge to a torsion free G_2 structures, and thus Ricci flat metric with holonomy G_2. I will start with the background of G_2 structure and the motivation of introducing the Laplacian flow, and then describe my recent progress on this flow (Joint work with Jason D. Lotay).
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Bruno Premoselli (Université Libre de Bruxelles)
Non-compactness of initial data sets in high dimensions.
We will discuss instability and non-compactness results for the conformal constraint equations in dimensions larger than six. These equations are obtained from the classical constraint equations of the evolution problem in General Relativity after a conformal change of variables. The system obtained in this way is a nonlinear critical system of elliptic PDEs.
We will show that in high dimensions blowing-up solutions of this system of equations naturally appear. The method is constructive, and the construction involves topological and blow-up techniques and relies on the 
a priori analysis of defects of compactness for critical elliptic PDEs. A reformulation in terms of the initial-value problem will be given.

15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Dmitri Panov (King's College London)
Circle-invariant definite connections and symplectic Fano 6-manifolds
The talk will be based on a joint work with Joel Fine. A definite connection on  a four manifold consists of a rank three Euclidean bundle with a metric connection whose curvature is maximally non-degenerate. I will explain why only the four sphere and the complex projective plane admit a definite connection with circle symmetry. The proof relies on properties of Hamiltonian S^1 actions on symplectic manifolds.
17:00 to 18:00 Welcome Wine Reception
Tuesday 28th June 2016
09:00 to 10:00 Lorenzo Foscolo (Stony Brook University)
ALC G2 manifolds
An ALC (asymptotically locally conical) manifold is a complete non-compact Riemannian manifold whose ends are modelled on circle fibrations over Riemannian cones of one lower dimension with fibres of asymptotically constant finite length. In dimension 4 and when the asymptotic cone is flat the acronym ALF (asymptotically locally flat) is more commonly used. (Conjectural) examples of ALC manifolds with G2 holonomy have appeared both in the physics and mathematics literature since the early 2000’s. ALC G2 manifolds provide interesting models for understanding how compact G2 manifolds can collapse to Calabi-Yau 3-folds. In this talk I will discuss the construction of various families of ALC G2 manifolds and describe their geometric properties. This is joint work with Mark Haskins (Imperial College London) and Johannes Nordström (University of Bath).
10:00 to 11:00 Andriy Haydys (Universität Bielefeld)
G2-instantons and Seiberg-Witten monopoles
I will show that both G2-instantons and the generalized Seiberg-Witten monopoles can degenerate to certain Fueter sections. This provides a basis for a (conjectural) relation between G2-injstantons and generalized Seiberg-Witten monopoles. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Thomas Walpuski (Massachusetts Institute of Technology)
G₂–instantons over twisted connected sums
In joint work with H. Sá Earp, I introduced a method to construct G₂–instantons over compact G₂–manifolds arising as the twisted connected sum of a matching pair of building blocks.  Using the fact that building blocks are K3 fibrations and work of Kuleshov on spherical bundles over K3s, this method can be used to produce concrete examples of G₂–instantons.  I will explain the abstract gluing theorem as well as one concrete example.  If time permits, I will also discuss the relation with work in progress (with A. Kovalev) on duality for building blocks, and/or the possibility of constructing singular G₂–instantons (with A. Jacob and H. Sá Earp).
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Simon Salamon (King's College London); (Politecnico di Torino)
Twistor analysis of a parabola
The Penrose twistor space CP^3 can be used to parametrize orthogonal complex structures on domains of R^4. In particular, a complex quadric Q defines a natural complex structure on R^4 minus a line. Quaternionic power series can then be lifted to holomorphic maps from Q to CP^3. Applying this to a quadratic polynomial allows one to understand the elementary geometry associated to a parabola and the circular cones in 3-dimensional space that contain it.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Yannick Herfray (University of Nottingham); (ENS - Lyon)
A 6d interpretation of 3d gravity
Wednesday 29th June 2016
09:00 to 10:00 Bobby Acharya (King's College London); (Abdus Salam International Centre for Theoretical Physics)
Geometry, Particle Physics and singular G2-manifolds
10:00 to 11:00 Arthur Lipstein (University of Durham)
On-Shell Diagrams for N = 8 Supergravity Amplitudes
Standard Feynman diagram techniques often obscure the underlying simplicity of on-shell scattering amplitudes. This difficulty can be overcome by working with on-shell diagrams, which do not contain virtual particles. In this talk, I will describe tree-level recursion relations for N = 8 supergravity amplitudes using a generalisation of the on-shell diagrams developed for planar N = 4 super-Yang-Mills, which imply new identities for non-planar diagrams as well as interesting relations between the N=4 and N=8 theories. If time permits, I will also speculate on the extension of this approach to loop-level. This talk is based on 1604.03046, in collaboration with Paul Heslop.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Radu Silviu Roiban (Pennsylvania State University)
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Kellogg Stelle (Imperial College London)
Black Holes and Higher Derivatives in Gravity
Quantum corrections to the gravitational action generically include quadratic terms in the curvature. Moreover, these terms are distinguished 
among other corrections in that their inclusion renders the theory renormalisable. The talk will discuss the changes their inclusion make to 
black hole solutions and other spherically symmetric solutions of the gravitational field equations.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 John Joseph Carrasco (CEA/Saclay)
Efficiency=Geometry? Decoding the DNA of prediction in gauge, gravity, and effective field theories.
One can easily be frustrated by the tremendous redundancy in possible physical description.  By this I mean the ability to choose a gauge, make field redefinitions, add any amount of auxiliary spectator matter, and the such.  Happily we can exploit such freedom to encourage the emergence of a new duality in gauge theories. The existence of a duality between color and kinematics exposes a hidden local double-copy structure inherent to prediction in many theories.  This structure weaves its way between theories both formal and phenomenological, from QCD to Gravity, from Chiral Perturbation Theory to Born-Infeld, and from open to closed superstring theories. The duality is sharpest at the level of the perturbative S-matrix — so I will focus my talk there, although I will also mention some recent provocative work beyond scattering.  I will mainly discuss challenges to generically achieving color-dual kinematic representations at the multi-loop level. I present a path forward that introduces, at least temporarily, a redundancy of description that we can exploit to map a set of functional relations to linear ones.  I will talk about this approach in terms of a geometric picture involving the graph of local graphs, discussing tradeoffs and applicability to long-standing problems.
17:15 to 17:45 Rutger Boels (Universität Hamburg)
Squaring relations from on-shell constraints
Thursday 30th June 2016
09:00 to 10:00 Emil Bjerrum-Bohr (Københavns Universitet (University of Copenhagen))
String Theory, Scattering Equations and Analytic Gluon Amplitudes
10:00 to 11:00 Ricardo Monteiro (CERN - European Organisation for Nuclear Research)
One Loop Integrands for Scattering Amplitudes from the Riemann Sphere
Worldsheet formulations of quantum field theories have had wide-ranging impact on the study of scattering amplitudes. However, the mathematical framework becomes very challenging on the higher-genus worldsheets required to describe loop effects. I will describe how, in such worldsheet models based on the scattering equations, formulae on genus one surfaces can be transformed to ones on the Riemann sphere. I will present formulae for the one-loop integrands of scattering amplitudes in gravity and gauge theory, with or without supersymmetry.

Related Links
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Yvonne Geyer (University of Oxford)
Two-loop Integrands from the Riemann Sphere
Co-authors: Lionel Mason (University of Oxford), Ricardo Monteiro (CERN), Piotr Tourkine (University of Cambridge)

Worldsheet formulations of quantum field theories have had wide-ranging impact on the study of scattering amplitudes. However, the mathematical framework becomes very challenging on the higher-genus worldsheets required to describe loop effects. I will describe how, in such worldsheet models based on the scattering equations, formulae on higher-genus surfaces can be transformed to ones on the Riemann sphere. My talk will focus on new developments at two loops for both supergravity and super Yang-Mills theory, and briefly discuss a proposal for the all-loop integrand.

Related Links
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Peter Goddard (Institute for Advanced Study, Princeton)
Solving the Scattering Equations
Describing work in collaboration with Louise Dolan, I will discuss the scattering equations, originally introduced in 1972 by Fairlie and Roberts searching for new dual models, rediscovered by Gross and Mende in 1988, discussing the high energy behaviour of string theory, and more recently shown by Cachazo, He and Yuan to provide a kinematic basis for describing remarkable formulae for tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons). We reformulate the scattering equations for N particles as a system of N -3 homogeneous polynomial equations in N - 2 complex variables, which are linear in each variable separately. The linearity of the equations enables their explicit solution in terms of the roots of a single-variable polynomial of degree (N-3)!, which can itself be explicitly constructed in terms of the Mandelstam variables formed from the momenta. The possible extension to one loop and the special case of four-dimensional space-time may also be briefly discussed.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Christopher White (University of Glasgow)
The classical double copy
The BCJ double copy relating perturbative scattering amplitudes in Yang-Mills and gravity theories has received a great deal of attention, and hints at a deeper, nonperturbative, relationship. This talk will review recent work on relating exact classical solutions in Yang-Mills theory and gravity, including familiar black holes as special cases. Furthermore, by considering radiating particles, I will provide strong evidence that this classical double copy is directly related to the BCJ story for amplitudes. 
19:30 to 22:00 Formal Dinner at Emmanuel College
Friday 1st July 2016
09:00 to 10:00 Oliver Schlotterer (Max-Planck-Institut für Gravitationphysik)
Universality in string interactions
Co-Authors: Yu-tin Huang (National Taiwan University), Congkao Wen (Tor Vergata Rome)

In this talk based on arXiv:1602.01674, we provide evidence for universality in the low-energy expansion of tree-level string interactions. More precisely, in the $\alpha'$-expansion of tree-level scattering amplitudes, we conjecture that the leading transcendental coefficient at each order in $\alpha'$ is universal for all perturbative string theories. We have checked this universality up to seven points and trace its origin to the ability to restructure the disk integrals of open bosonic string into those of the superstring. The accompanying kinematic functions have the same low-energy limit and do not introduce any transcendental numbers in their $\alpha'$-corrections. Universality in the closed-string sector then follows from the Kawai-Lewellen-Tye-relations.
10:00 to 11:00 Nathan Berkovits (ICTP-SAIFR/IFT-UNESP, Sao Paulo)
Untwisting the pure spinor formalism
The pure spinor formalism for the superstring  is a twisted N=2 superconformal field theory where the BRST current and the b ghost are the two fermionic generators. After untwisting this N=2 superconformal field theory and defining the N=1 superconformal generator G = j_{BRST} + b, the pure spinor formalism can be described with manifest N=1 worldsheet supersymmetry. This N=1 supersymmetric description is useful for comparing with the RNS formalism and for constructing worldsheet supersymmetric sigma models for Ramond-Ramond backgrounds such as AdS_5xS^5.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Piotr Tourkine (University of Cambridge)
On the null string origin of the ambitwistor string
 Co-Author: Eduardo Casali (Math. Institute, Oxford)
 Based on: [arXiv:1606.05636]

  The CHY formulae are a set of remarkable formulae describing the scattering amplitudes of a variety of massless theories, as worldsheet integrals localized on the solutions to certain polynomial equations (scattering equations). These formulae arise from a new class of holomorphic string theories called Ambitwistor strings that encode exactly the classical dynamics of the supergravity (Yang-Mills) modes of string theory.

 Despite some recent progress by W. Siegel and collaborators, it remained as an open question as to what extent this theory was connected to the full string theory. The most mysterious point being certainly that the localization equations of the ambitwistor string also appear in the zero tension limit of string theory (alpha’ to infinity), which is the opposite limit than the supergravity one (alpha’ to zero).

 In this talk, I’ll report on some work in collaboration with E. Casali and argue that the ambitwistor string is actually a tensionless string. Using some forgotten results on the quantization of these objects, we explain that the quantization of tensionless strings is ambiguous, and can lead either to a ``higher-spin like'' theory, or to the ambitwistor string, hence clarifying the previously mentioned paradox. In passing, we see that the degenerations of the tensile worldsheet that lead to tensionless strings make connection with Galilean Conformal Algebras and the (3d) BMS algebra.

12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Alexander Ochirov (University of Edinburgh)
Color-kinematics duality for QCD and pure gravities
Color-kinematics duality has been shown to be present in QCD scattering amplitudes with massive quarks. More generally, the duality was extended to gauge theories with flavored massive matter in any Lie-algebra representation, with the resulting double copy for gravitational amplitudes. In this talk, I will discuss the details of the extension, including a new decomposition of QCD amplitudes into independent color-ordered primitives, and how the duality imposes new relations between the primitives. Furthermore, I will discuss how the extension of the duality can be used to compute gravitational amplitudes with matter and how this can be used to obtain amplitudes in pure gravity theories.

Associated publications together with Henrik Johansson (Uppsala University & Nordita):
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Jaroslav Trnka (University of California, Davis)
Singularity structure of gravity amplitudes
Coauthor: Enrico Herrmann

There has been a lot of progress in exploring new mathematical structures in N=4 SYM amplitudes: on-shell diagrams and Amplituhedron in the planar sector, and special properties of integrands beyond the planar limit. I will review some of these developments and then focus how the similar ideas can be extended to N=8 supergravity amplitudes. As it turns out the singularity structure of supergravity amplitudes is very different and it exhibits some surprising features at both UV and IR.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons