It has been conjectured that the fundamental theory of strings and branes has an $E_{11}$ symmetry. I will explain how this conjecture leads to a generalised space-time, which is automatically equipped with its own geometry, as well as equations of motion for the fields that live on this generalised space-time.

# Past String Theory Seminar

Form factors form a bridge between the purely on-shell amplitudes and the purely off-shell correlation functions. In this talk, we study the form factors of general gauge-invariant local composite operators in N=4 SYM theory via on-shell methods. At tree-level and for a minimalnumber of external fields, the form factor exactly realises the spin-chain picture of N=4 SYM theory in the language of scattering amplitudes. Via generalised unitarity, we obtain the cut-constructible part of the one-loop correction to the minimal form factor of a generic operator. Its UV divergence yields the complete one-loop dilatation operator of the theory. At two-loop order, we employ unitarity to calculate the minimal form factors and thereby the dilatation operator for the Konishi primary operator and all operators in the SU(2) sector. For the former operator as well as other non-protected operators, important subtleties arise which require an extension of the method of unitarity.

Recently, as part of a project to find CY manifolds for which both the Hodge numbers (h^{11}, h^{21}) are small, manifolds have been found with Hodge numbers (4,1) and (1,1). The one-dimensional special geometries of their complex structures are more complicated than those previously studied. I will review these, emphasising the role of the fundamental period and Picard-Fuchs equation. Two arithmetic aspects arise: the first is the role of \zeta(3) in the monodromy matrices and the second is the fact, perhaps natural to a number theorist, that through a study of the CY manifolds over finite fields, modular functions can be associated to the singular manifolds of the family. This is a report on joint work with Volker Braun, Xenia de la Ossa and Duco van Straten.

We propose a non-perturbative formulation of structure constants of single trace operators in planar *N=4* SYM. We match our results with both weak and strong coupling data available in the literature. Based on work with Benjamin Basso and Pedro Vieira.

I shall demonstrate, under some mild assumptions, that the symmetry group of extreme, Killing, supergravity horzions contains an sl(2, R) subalgebra. The proof requires a generalization of the Lichnerowicz theorem for non-metric connections. The techniques developed can also be applied in the classification

of AdS and Minkowski flux backgrounds.