Past Relativity Seminar

I shall report on recent progress, in Australia and Germany, on the empirical evaluation of special values of L-functions by minors of period matrices whose elements include Feynman integrals from diagrams with up to 20 loops. Previously such relations were known only for diagrams with up to 6 loops.
 

An essential ingredient of AdS/CFT, dS/CFT and other dualities is a geometric notion of scattering that refers to asymptotics rather than, say, infinite time limits. Though one expects non-perturbative versions to exist in the case of linear quantum fields (and non-linear classical fields), this has been rigorously implemented in Lorentzian settings only relatively recently. The goal of this talk will be to give an overview in different geometrical setups, including asymptotically Minkowski, de Sitter and Anti-de Sitter spacetimes. In particular I will discuss recent results on classical scattering and particle interpretations, compare them with the setup of conformal scattering and explain how they can be used to construct "in-out" Feynman propagators (based on joint works with Christian Gérard and András Vasy).

Amplitudes in planar N=4 SYM are dual to light-like polygonal Wilson-loop expectation values. In many cases their perturbative expansion can be expressed in terms of multiple polylogarithms which also obey certain single-valuedness conditions or branch cut restrictions. The rigidity of this function space, together with a few other conditions, allows one to construct the six-point amplitude -- or hexagonal Wilson loop -- through at least five loops, and the seven-point amplitude through 3.5 loops. Then one can "fold" the polygonal Wilson loops into multiple degenerate configurations, expressing the limiting behavior in terms of simpler function spaces, and learning in the process about how amplitudes factorize.
 

After a brief review of holographic features of general relativity in 3 and 4 dimensions, I will show how to derive the transformation laws of the Bondi mass and angular momentum aspects under finite supertranslations, superrotations and complex Weyl rescalings.
 

The Feynman diagram expansion of scattering amplitudes in perturbative superstring theory can be written (for closed strings) as a series of integrals over compactified moduli spaces of Riemann surfaces with marked points, indexed by the genus. Therefore in genus 0 it is reasonable to find, as it often happens in QFT computations, periods of M_{0,N}, which are known to be multiple zeta values. In this talk I want to report on recent advances in the genus 1 amplitude, which are related to the development of 2 different generalizations of classical multiple zeta values, namely elliptic multiple zeta values and conical sums.

In this talk I will present a class of super-Wilson loops in N=4 super Yang-Mills theory. The expectation value of these operators has been shown previously to be invariant under a Yangian symmetry. I will show how the kinematics of such super-Wilson loops can be described in a twistorial way and how this leads to compact, manifestly super-conformal invariant expressions for some two-point functions.
 

I will survey of some of the many significant connections between integrable many-body physics and mathematics. I exploit an algebraic structure called a fusion category, familiar from the study of conformal field theory, topological quantum field theory and knot invariants. Rewriting statistical-mechanical models in terms of a fusion category allows the derivation of combinatorial identities for the Tutte polynomial, the analysis of discrete ``holomorphic'' observables in probability, and to defining topological defects in lattice models. I will give a little more detail on topological defects, explaining how they allows exact computations of conformal-field-theory quantities directly on the lattice, as well as a greatly generalised set of duality transformations.
 

I review work done in collaboration with Siegel and Zwiebach,  in which a doubled geometry is developed that provides a spacetime  action containing the standard gravity theory for graviton, Kalb-Ramond field and dilaton plus higher-derivative corrections. In this framework the T-duality O(d,d) invariance is manifest and exact to all orders in $\alpha'$.  This theory by itself does not correspond to a standard string theory, but it does encode the Green-Schwarz deformation characteristic of heterotic string theory  to first order in $\alpha'$ and a Riemann-cube correction to second order in  $\alpha'$. I outline how this theory may be extended to include arbitrary string theories. 

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



Despite some recent progress by W. Siegel and collaborators, it remains 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 progress with E. Casali (Math. Inst. Oxford) 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 theory, or to the ambitwistor string, hence solving 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.

Loop computations put the 'quantum' into quantum field theory. Much effort has focused on their structure and properties, with most spectacular progress in maximally supersymmetric gauge theories in the planar limit. These theories are however quite far from reality as described for instance in the standard model of particle physics. In this talk I'll report on ongoing work using BCFW on-shell recursion to obtain loop amplitude integrands in a much more realistic theory, pure Yang-Mills theory, using methods which apply directly to the standard model.

Black holes are one of the few available laboratories for testing theoretical ideas in fundamental physics. Since Hawking's result that they radiate a thermal spectrum, black holes have been regarded as thermodynamic objects with associated temperature, entropy, etc. While this is an extremely beautiful picture it has also lead to numerous puzzles. In this talk I will describe the two-loop correction to scalar correlation functions due to \phi^{^4} interactions and explain why this might have implications for our current view of semi-classical black holes.
 

I will summarize recent (arXiv:1603.05262) and upcoming work with Igor Buchberger and Oliver Schlotterer. We construct a map from n-point 1-loop string amplitudes in maximal supersymmetry to n-3-point 1-loop amplitudes in minimal supersymmetry. I will outline a few implications for the quantum string effective action.

In this talk, a concrete realization of the Bern-Carrasco-Johansson (BCJ) duality between color and kinematics in non-abelian gauge theories is presented. The method of Berends-Giele to package Feynman diagrams into currents is shown to yield classical solutions to the non-linear Yang-Mills equations. We describe a non-linear gauge transformation of these perturbiner solutions which reorganize the cubic-diagram content such that the kinematic dependence obeys the same Jacobi identities as the accompanying color factors. The resulting tree-level subdiagrams are assembled to kinematic numerators of tree-level and one-loop amplitudes which satisfy the BCJ duality.

 In the last few years, it has become clear that there are striking connections between supersymmetry and geometric representation theory.  In this talk, I will discuss boundary conditions in three dimensional gauge theories with N = 4 supersymmetry.  I will then outline a physical understanding of a remarkable conjecture in representation theory known as `symplectic duality.

In this talk I will present some recent work on the amplituhedron formulation of scattering amplitudes. Very recently it has been conjectured that amplitudes in planar N=4 sYM are nothing else but the volume of a completely new mathematical object, called amplituhedron, which generalises the positive Grassmannian. After a review of the main ingredients which will be used, I will discuss some of the questions which remain open in this framework. I will then describe a new direction which promises to solve these issues and compute the volume of the amplituhedron at tree level.

 

Modular invariance is ubiquitous in string theory.   This is the symmetry of genus-one amplitudes, as well as the non-perturbative duality symmetry of type IIb superstring in ten dimensions.  The alpha’ expansion of string theory amplitudes leads to interesting new modular forms. In this talk we will describe the properties of the new modular forms. We will explain that the modular forms entering the alpha’ expansion of genus one type-II superstring amplitude are naturally expressed as particular values of single valued elliptic multiple polylogarithm.  They are natural modular generalization of the single valued elliptic multiple-zeta introduced by Francis Brown. 

Recent results showed that the low energy expansion of closed superstring amplitudes can be expressed in terms of

single-valued multiple elliptic polylogarithms. I will explain how these functions may be defined as iterated integrals on the torus and

sketch how they arise from Feynman integrals.

 About 10 years ago Minahan and Zarembo made a remarkable discovery: the one-loop Dilatation Operator in the SO(6) sector of planar N=4 SYM can be identified with the Hamiltonian of an integrable spin chain. This one-loop Dilatation operator was obtained by computing a two-point correlation function at one loop, which is a completely off-shell quantity. Around the same time, Witten proposed a duality between N=4 SYM and twistor string theory, which initiated a revolution in the field of on-shell objects like scattering amplitudes. In this talk we illustrate that these techniques that have been sucessfully used for on-shell quantities can also be employed for the computation of off-shell quantities by computing the one-loop Dilatation Operator in the SO(6) sector. The first half of the talk will be dedicated to doing this calculation using MHV diagrams and the second half of the talk shows the computation in twistor space. 

These two short talks will be followed by an informal afternoon session for those interested in further details of these approaches, and in form factors in Class Room C2 from 2-4.30 pm then from 4.30pm in N3.12.  All are welcome.

I will describe recent progress in describing 4-dimensional N = 8 supergravity using the framework of ambitwistor string theory.

New ambitwistor string models are presented for a variety of theories and older models are shown to work at 1 loop and perhaps higher using a simpler formulation on the Riemann sphere.

The study of infrared singularities, due to the emission of “soft” (low momentum) gauge bosons, remains a highly active research area in a variety of quantum field theories. After motivating both phenomenological and formal reasons as to why we should care about IR singularities, this talk will review their structure in QED, QCD and quantum gravity, examining the similarities and differences between these three contexts. The role of Wilson lines will be examined, which provide a useful unifying language. Finally, I will examine recent work on moving beyond the soft approximation, and why this might be useful.

A systematic understanding of the low energy limit of string theory scattering amplitudes is essential for conceptual and practical reasons. In this talk, I shall report on a work where this limit has been analyzed using tropical geometry. Our result is that the field theory amplitudes arising in the low energy limit of string theory are written in a very compact form as integrals over a single object, the tropical moduli space. This picture provides a general framework where the different aspects of the low energy limit of string theory scattering amplitudes are systematically encompassed; the Feynman graph structure and the ultraviolet regulation mechanism. I shall then give examples of application of the formalism, in particular at genus two, and discuss open issues.

No knowledge of tropical geometry will be assumed and the topic shall be introduced during the talk.

I will present a systematic approach to two-dimensional N=(2,2) supersymmetric gauge theories in curved space, with a particular focus on the two-dimensional Omega deformation. I will explain how to compute Omega-deformed A-type topological correlation functions in purely field theoretic terms (i.e. without relying on a target-space picture), improving on previous techniques. The resulting general formula simplifies previous results in the Abelian (toric) case, while it leads to new results for non-Abelian GLSMs.

In this talk, I develop the Hopf algebra of renormalization, as established by Connes and Kreimer. I then use the correspondence between commutative Hopf algebras and affine groups to show that the energy scale dependence of the renormalized theory can be expressed as a Maurer Cartan connection on the renormalization group.

In this talk I am going to discuss our recent and on-going work on an integral representation of tree-level S-matrices for massless particles. Starting from the formula for gravity amplitudes, I will introduce three operations acting on the integrand that produce compact and closed formulas for amplitudes in various other theories of massless bosons. In particular these includes Yang-Mills coupled to gravity, (Dirac)-Born-Infeld, U(N) non-linear sigma model, and Galileon theory. The main references are arXiv:1409.8256, arXiv:1412.3479.

A recurring theme in attempts to understand the quantum theory of gravity is the idea of "Gravity as the square of Yang-Mills". In recent years this idea has been met with renewed energy, principally driven by a string of discoveries uncovering intriguing and powerful identities relating gravity and gauge scattering amplitudes. In an effort to develop this program further, we explore the relationship between both the global and local symmetries of (super)gravity and those of (super) Yang-Mills theories squared. 



In the context of global symmetries we begin by giving a unified description of D=3 super-Yang-Mills theory with N=1, 2, 4, 8 supersymmeties in terms of the four division algebras: reals, complex, quaternions and octonions. On taking the product of these multiplets we obtain a set of D=3 supergravity theories with global symmetries (U-dualities) belonging to the Freudenthal magic square: “division algebras squared” = “Yang-Mills squared”! By generalising to D=3,4,6,10 we uncover a magic pyramid of Lie algebras.



We then turn our attention to local symmetries. Regarding gravity as the convolution of left and right Yang-Mills theories together with a spectator scalar field in the bi-adjoint representation, we derive in linearised approximation the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincaré. As a concrete example we focus on the new-minimal (12+12, N=1) off-shell version four-dimensional supergravity obtained by tensoring the off-shell (super) Yang-Mills multiplets (4+4, N =1) and (3+0, N =0).

This talk describes a method to compute supersymmetric tree amplitudes and loop integrands in ten-dimensional super Yang-Mills theory. It relies on the constructive interplay between their cubic graph organization and BRST invariance of the underlying pure spinor superspace description. After a general introduction to this kind of superspace, we discuss a canonical set of multiparticle building blocks which represent tree level subdiagrams and are guided by their BRST transformation. These building blocks are shown to yield a compact solution for tree level amplitudes, and the applicability of the BRST approach to loop integrands is exemplified through recent examples at one-loop.

This is another opportunity to hear the 2013 LMS Presidential Address:

Abstract: The idea of space is central to the way we think.  It is the technology we have evolved for interpreting our experience of the world.  But space is presumably a human creation, and even inside mathematics it plays a variety of different roles, some modelling our intuition very closely and some seeming almost magical.  I shall point out how the homotopy category in particular breaks away from its own roots.  Then I shall describe how quantum theory leads us beyond the well-established notion of a topological space into the realm of noncommutative geometry.  One might think that noncommutative spaces are not very space-like, and yet it is noncommutativity that makes the world look as it does to us, as a collection of point particles.

We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space–time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for tree– level scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories. 

According to Witten a gauge theory with a mass gap contains a possibly trivial Topological Field Theory  (TFT) in the infrared.  We show that in SU(N) YM it there exists a trivial TFT defined by   twistor Wilson loops whose v.e.v. is 1 in the large-N limit for any shape of the loops supported on certain Lagrangian submanifolds of space-time that lift to Lagrangian submanifolds of twistor space.

We derive a new version of the Makeenko-Migdal loop equation for the topological twistor Wilson loops, the holomorphic loop equation, that involves the change of variables in the YM functional integral from the connection to the anti-selfdual part of the curvature and the choice of a holomorphic gauge.

Employing the holomorphic loop equation and viewing Floer homology the other way around,
we associate to arcs asymptotic in both directions to the cusps of the Lagrangian submanifolds the critical points of an effective action implied by the holomorphic loop equation. The critical points of the effective action, being associated to the homology of the punctured Lagrangian submanifolds, consist of surface operators of the YM theory, supported on the punctures.  The correlators of surface operators in the TFT satisfy for large momentum the constraint that follows by the renormalization group and by the asymptotic freedom and they are saturated by an infinite sum of pure poles of scalar and pseudoscalar glueballs, whose joint spectrum is exactly linear in the mass squared.

For several physical purposes we outline  a related construction of a twistorial Topological String Theory dual to the TFT, that involves the Chern-Simons action on Lagrangian submanifolds of  
twistor space.