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.

# Past Relativity Seminar

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.