Dr Benjamin Doyon (Integrable/conformal field theory; random loops; systems out of equilibrium.)

Title:
Quantum field theory out of equilibrium

Abstract:
This project will develop various notions of quantum field theory
out of equilibrium. It will concentrate on steady states out of equilibrium,
where there is a steady transfer of charge or energy from one region of
space to another other, due to a difference of electric potential or of
temperature between the two regions. The real-time Keldysh formulation
will be studied, as well as how it leads to the steady-state density matrix;
the proof of this connection will be developed in simple models (free fields).
Using the steady state density matrix, various physical quantities (current,
correlations in space or time) will be calculated again in simple models.
Students who are able to get further can follow a variety of routes; for instance,
calculating all current fluctuations, or developing the perturbation theory
with respect to a free field. Some of the basic ideas can be found in
http://arxiv.org/abs/1202.0239.

Prerequisite:
AQFT


Dr Nadav Drukker

Title:
Wilson loops in AdS/CFT

Abstract:
This project will focus on the AdS/CFT correspondence and calculate in
it one of the simplest nontrivial observables: A circular Wilson loop.
The calculation has three parts: A field theory calculation, a matrix
model calculation (combinatorics) and a classical string calculation.
This project will review all three with the purpose of deriving a result
from the gauge theory in perturbation theory, re-sum the full perturbative
expansion (employing the matrix model), extrapolating the result to
strong coupling and recovering the result of a minimal string solution in
anti de Sitter space. For ambitions students there are natural
generalizations to calculate more involved observables.

Prerequisites:
GR, QFT, AQFT, String Theory


Dr Nikolay Gromov (Gauge theories; AdS/CFT correspondence; Integrability.)

Project 1

Title:
Integrability and Sin-Gordon Theory

Abstract:
The Sin-Gordon theory is a field theory in 2D with many nice features.
The classical equations of motion  are integrable, which means that many 
nice solutions can be constructed explicitly (see this for many nice
examples). For the project it is proposed to study this nice theory at 
the quantum level using integrability.

Prerequisites:
Field theory


Project 2

Title:
Partition function of Hubbard model

Abstract:
The Hubbard model is an integrable theory of electrons jumping between a
nodes of 1+1D Cristal. It is known to be integrable and the quantum spectrum
is given by Bethe equations. Some variant of this system describes Graphen
(Nobel prize 2010). Interestingly it is also closely related to the
AdS/CFT and  Super Yang-Mills in 4D! 

For the project it is proposed to study the partition function for this
model by means of the Thermodynamic Bethe Ansatz and Y-system (i.e. methods
very important in AdS/CFT).

Prerequisites:
General Thermodynamics


Project 3

Title:
BFKL Limit and AdS/CFT correspondence

Abstract:
BFKL Limit is a real physical limit in QCD describing the Deep Inelastic
Scattering. The same limit can be studied in the framework of AdS/CFT
and should actually coincide with the realistic QCD results. 

Prerequisites:
Field theory


Project 4

Nonplanar corrections to matrix models: topological recursion

For the project it is proposed to study some simple matrix model
and apply a procedure known as a topological recursion to find its non-planar corrections.
Mathematica will be used for this project to simplify some calculations.


Prof George Papadopoulos

Title: Lichnerowicz Theorem and Supersymmetry


Abstract:  The classical Lichnerowicz Theorem relates the zero modes of the
Dirac operator on a closed manifold to the parallel spinors. This can be used
together with the index theorem to provide a topological formula for the number
of parallel Killing spinors in a supergravity theory.  The project will involve
the derivation of the Lichnerowicz Theorem as well as the exploration of the
topological formula for the parallel spinors.  Generalization of the  Lichnerowicz
Theorem may also be investigated in the context of black hole horizons.

Prerequisites:
This project requires familiarity
with Manifolds, Supersymmetry, General Relativity, and some algebraic topology.


Dr Andreas Recknagel

1. Matrix factorisations and boundary conditions for supersymmetric 
  Landau-Ginzburg models

Matrix factorisations are relations of the form 
E J = J E = W 1_N 
where W is a polynomial, E and J are NxN matrices with poynomial entries,
and 1_N is the NxN unit matrix. 
In hep-th/0305133 it was shown that matrix factorisation describe N=2
supersymmetric boundary condition for Landau-Ginzburg models in two dimensions.
The project should first of all review this derivation and discuss basic
general constructions of matrix factorisations along with simple examples. 
Taking the relation between Landau-Ginzburg models and superconformal
field theories as a given, the project should also discuss topologically
twisted N=2 SCFTs and compare their boundary spectrum to the cohomologies
of matrix factorisations. This will in particular require to understand
N=2 SCFTs and some elements of boundary CFT. 

Prerequisites: conformal field theory


2. Boundary conformal field theory and D-branes

The project is similar in scope to the one offered by Dr Gerard Watts,
with greater emphasis on string theory applications. 
Based on a good understanding of conformal field theory on the plane,
thefollowing notions need to be developed: conformal boundary conditions,
gluing conditions for Virasoro and other symmetry generators, implementation
via boundary states, boundary fields (corresponding to open string vertexoperators),
non-linear constraints on structure constants (in particular the Cardy conditions). 
Applications to CFTs relevant for string theory should be discussed,
in particular free bosons (Dirichlet and Neumann boundary conditions, rotated 
branes), ideally also Gepner models (superstring compactifications). 

Prerequisites: conformal field theory and string theory


Dr Sakura Schafer-Nameki

Project 1

Title:
The More Minimal Supersymmetric Standard Model

This project is about exploring the More Minimal Supersymmetric Standard 
Model or so-called Inverted Hierarchy Models, which are particular types 
of supersymmetric extensions of the Standard Model, where the first and 
second generation particles are heavy and the third generation is light. 
There will be theoretical part, exploring the main features of these 
models, e.g.

http://arxiv.org/abs/hep-ph/9607394

and there will be an applied part, where computations of the spectra, 
decays and phenomenology of these models will be explored using simple 
computer programs, such as softsusy (as introduced in the SUSY lecture).

Prerequisits: Supersymmetry course, QFT, AQFT.



Dr Gerard Watts

Title:
Boundary conformal field theory

Abstract:
To construct quantum conformal field theories on domains with boundaries 
one has to understand the nature of boundary conditions on a quantum 
field theory. For a conformal field theory, a boundary condition can be 
encoded in a special "boundary state". The project will investigate the 
nature of boundary conditions and the fields that are defined on the 
boundary; the renormalisation group flows and the algebraic 
characterisation of conformal boundary conditions, the renormalisation 
group fixed points. The project should cover an advanced topic such as 
the conformal boundary conditions on a free boson; the integrable 
description of boundary renormalisation group flows; the classification 
of boundary conditions for Virasoro minimal models; the boundary 
conditions of Liouville theory.

Prerequisites:
Conformal field theory


Prof Peter West

Project 1

Title:
Kac-Moody Algebras and Supergravity Theories

Abstract:
The main aim of this project is to write an account of the Kac-Moody algebras
discovered in 1969. This begins with a proper understanding of the semi-simple
Lie algebras as classified by Cartan and Killing, in particular giving their
formulation in the Serre presentation. This then allows a straightforward
definition of Kac-Moody algebras. The project could then give an account of
Lorentzian Kac-Moody algebras and work out, at low levels, the particular
Kac-Moody algebra and some of its representations. 

Finally, the project may use the theory of non-linear realisations to apply
Kac-Moody algebras to construct supergravity theories. 

Prerequisites:
A good knowledge of Lie algebras is required and perhaps some knowledge of
supersymmetry if the latter topics are to be included.