Forty-Second Meeting of the North British Mathematical Physics Seminar

Entanglement entropy is a useful measure of entanglement, a quintessentially quantum feature of physical systems. Though its intricate nature renders it hard to calculate in all but the simplest settings, for strongly coupled field theories the tools of holography come to the rescue. This talk will review the proposal of holographic entanglement entropy in AdS/CFT for general time-dependent setting (for which the requirement of general covariance provided a useful guiding principle), focussing on the recently-verified consistency of this prescription with field theory causality.

Holographic entanglement entropy is a prescription relating a quantum idea, entanglement, to a geometric one, a minimal surface. For consistency, a constraint on the topology of the admissible class of minimal surfaces is required, but so far this has been applied ad hoc, with no derivation from more basic principles. I'll review recent progress towards a proof of the minimal surface proposal, before describing how the topology fits into the picture.

The duality proposed by Bern, Carrasco and Johansson (BCJ) suggests a cubic formulation of the Yang-Mills and gravity amplitude. A rather interesting feature of this formulation is that the kinematic dependence of the amplitudes presents algebraic properties similar to the colour dependence of the Yang-Mills amplitude. I will discuss about a systematic construction of the building blocks of BCJ relation and new formulations of Yang-Mills and gravity amplitudes suggested by the duality.

We consider the calculation of scattering amplitudes in field theories dual to Lifshitz spacetimes. These amplitudes provide an interesting probe of the IR structure of the field theory; our aim is to use them to explore the observable consequences of the singularity in the spacetime. We assume the amplitudes can be related by T-duality to a Wilson loop, as in the AdS case, and determine the bulk minimal surfaces for the simplest cusp Wilson loop. We use this to determine the leading IR singularity in the amplitude.

Non-Linear Integral Equations (NLIEs) were introduced in the 1990's as a powerful tool to study the scaling limit of integrable lattice models. Generically, the scaling theory describing the critical point of a spin chain is a Conformal Field Theory (CFT) with a discrete spectrum of scaling dimensions, which can be extracted from the NLIEs by a well-controlled procedure. In this talk, I will present a simple integrable spin-chain model whose Bethe Ansatz equations are governed by a singular kernel: I will explain how to derive the NLIEs in this situation, and how to treat the singularity to obtain the CFT spectrum. It turns out that the corresponding CFT is the SL(2,R)/U(1) "black hole" WZW model, a toy model of CFT with non-compact target space.

One dimensional Heisenberg spin chains can be used to model certain quasi one dimensional materials. Using the vertex operator approach due to Jimbo and Miwa, it is possible to compute exact results for correlation functions of the spin 1/2 XXZ chain and so calculate the dynamic structure factors of these materials objects measurable in inelastic neutron scattering experiments. I hope to give an overview of the techniques involved and briefly talk about our application of this approach in the spin 1 case, the goal of which being to compute exact form factors.