### Lecture note on special topics @ Imperial College 2013

Collection of scanned taken notes of special topics courses at Imperial College London in 2013. They are basically short (3-4 lectures) courses given after the final exams for theoretical physics master students. Since I had these interesting stuffs scanned, I will put them here just in case someone will find them useful.

Well… there are several mistakes in these files which are mostly misspelling and scanning. If you found some pages missed in the first file, I probably tried to rescanning those missing pages in the second one. Also, there are probably a lot of stuff that I misunderstood what the lecturer meant. So for those who want to learn something from these notes, please be very careful. I really apologise for that.

Abstracts of the courses here are taken from the courses descriptions in the QFFF website. They update the page every year for new courses and hardly keep any old courses descriptions. So I hope it is ok for me to copy them here, just for the record.

I thanks Anne-Sylvie Deutsch for letting me scanned last part of her taken notes on Causal set QFT.

– **Supersymmetry on curved space and Localisation** (Stafano Cremonesi) Localisation_1, Localisation_2

Computing path integrals of quantum field theories is a notably hard problem: we all know how to make perturbative approximations, but these are of little use if interactions are strong. Over the last five years, a technique called localisation has been developed to solve path integrals exactly for large classes of interacting supersymmetric field theories on curved spaces.

The aim of the course is to give an introduction to these recent developments. I will start by reviewing the idea of localisation in the context of ordinary integrals. I will then explain how supersymmetric field theories can be defined on curved space(time)s. Finally, I will introduce localisation for such supersymmetric field theories and explain how it reduces path integrals to finite-dimensional integrals that can be more easily evaluated.

*If you happens to scan through my scribble and somehow think that it is really interesting, or happens to work on localisation-related subject, I would wholeheartedly recommend you to take a look at the note on this topics that Stefano typeset here*

– **Causal Set QFT** (Leron Borsten) Causal Set_1, Causal Set_2

Causal set theory is an approach to the problem of quantum gravity founded upon two intuitive and intimately related conceptual principles regarding the microscopic nature of spacetime. First, the quantum structure of spacetime is fundamentally discrete; the “continuum” does not enter in its definition. Second, Êthe causal ordering of spacetime events is of primary importance. These principles are encapsulated in the notion of “casual sets” – the basic objects of causal set theory, which replace the spacetime manifolds of GR. However, in abandoning the continuum we must forgo many of our most trusted concepts and mathematical tools. In particular, without these familiar aids it is not clear a priori how to consistently formulate quantum field theory on a causal set without recourse to the continuum.

In these lectures we will motivate and introduce the basic concepts of causal set theory in the context of quantum gravity. Using these ideas we will give an intrinsic casual set definition of a propagator for a quantum scalar field, making no reference to the continuum. By demonstrating how to build causal sets which are well approximated by continuous Lorentzian manifolds, we will show that these “causet propagators” are also well approximated by the propagators of conventional scalar quantum field theory (SQFT). (A classical analogy is that our molecular understanding of water is well approximated by fluid dynamics.) These constructions will be brought together to define SQFT on a causal set, which will be checked against its continuum SQFT approximation. Finally, we will explore what can be learnt about conventional continuum QFT from this causal set perspective.

– **Generalised Geometry** (Danial Waldram) Generalised Geometry

Generalised geometry refers to a set of particular extensions of conventional differential geometry. First introduced by mathematicians to describe generalisation of complex and symplectic geometry, it turns out that it is naturally adapted to give a unified description of supergravity theories, in a way that echoes the duality symmetries of string theory. In these lectures we introduce the basic notions of generalised geometry and study the simplest example: showing how they can be used to reformulate type II supergravity in ten-dimensions. This, for example, unifies all the NS-NS bosonic equation of motion into the vanishing of a generalised Ricci curvature tensor.

– ** Lattice field theory in an out of equilibrium** (David Weir) non perturbative QFT

Most of the quantum field theory taught in the QFFF MSc is perturbative. However, many of the big unsolved problems for which QFT is the right tool are very difficult, nonperturbative questions. These problems include the formation of the quark-gluon plasma and confinement in SU(N). Studying quantum field theories on a euclidean lattice provides natural UV and IR cutoffs, and there exist simulation algorithms that permit the calculation of correlation functions and other observables on these lattices.

In these special topics lectures, the theoretical underpinnings of lattice field theory and some exact results will be discussed, such as the LŸscher-Weisz solution. We will also discuss the formulation of gauge fields and, if time allows, fermions. Some time will be spent discussing simulation algorithms and their applicability. The final lecture will be a workshop, in which a provided simulation code will be used to generate and analyse results.

– **AdS/CFT** (Michela Petrini) AdSCFT_1, AdSCFT_2

In the first part of the course, after a brief introduction to D-branes in type II theories, I will discuss in some detail the AdS/correspondence for N=4 SYM in four dimensions. Then I will give an overview of how the correspondence can be generalised to confining theories, non supersymmetric ones and its applications to condensed matter systems.