Quantum Fields and Strings

We establish a direct connection between scattering amplitudes for bi-adjoint scalar theories and a classic polytope--the "associahedron"--known to mathematicians since the 1960s. We find an associahedron naturally living in kinematic space. The tree level scattering amplitude is simply a geometric invariant of the associahedron called its "canonical form" [2], which is a differential form on kinematic space with logarithmic singularities on the boundaries of the associahedron.

Known N=4 theories in four dimensions are characterized by a choice of gauge group, and in some cases some "discrete theta angles", as classified by Aharony, Seiberg and Tachikawa. I will review how this data, for the theories with algebra su(N), is encoded in various familiar realizations of the theory, in particular in the holographic AdS_5 \times S^5 dual and in the compactification of the (2,0) A_N theory on T^2.

We review recent developments concerning the dynamics of QCD in 2+1 dimensions. We will discuss the phases of the theory depending on the matter representations and the Chern-Simons level. We present several new dualities and conjectures about the behaviour of these theories in their strongly coupled phases.

A state is called a Markov state if it fulfil the important condition of saturating the Strong Subadditivity inequality. I will show how the vacuum state of any relativistic QFT is a Markov state when reduced to certain geometric regions of spacetime. A characterisation of this regions will be presented as well as two independent proofs of the Markov condition in QFT.