Structure and Symmetry

Abstract: Anomalies in various physical theories have been understood from the relatively new perspective of functorial field theories, which provide a rigorous approach to quantum field theory, and have broad applications to string theory (for instance, in the 6-dimensional (2,0)-theory) and to condensed matter physics (for instance, in symmetry-protected topological phases of matter). This project will study various topical features of anomalies and other field theoretic phenomena from the functorial perspective in this rapidly-evolving field.

Abstract: Exact  expressions  for boundary  states  of  integrable quantum spin
chains  (eigenstates  of  the   Hamiltonian  acting  parallel  to the
boundary) can sometimes  be obtained via the  vertex operator approach
to solvable lattice models developed  by Jimbo, Miwa and collaborators
in the  1990s. This  approach relies on  the representation theory of
quantum  affine algebras.   In this  project you  will construct such
boundary  states in  a new  class of  models -  those associated with
'twisted Yangian' algebraic symmetries.   You will analyse the scaling
limit of these  models and compare your results  with calculations for
the corresponding  boundary scattering matrices of  integrable quantum
field  theories. You  will then  use your  boundary states  in order  to
construct exact  expressions for  correlation functions  in integrable
quantum spin chains with open boundaries.

Abstract: See the link below.

Abstract: T-duality in string theory has revealed the existence of consistent string backgrounds which have no description in terms of conventional Riemannian geometry. These backgrounds can be defined in the framework of double field theory, which however is not understood globally yet. A promising approach to a global formulation of double field theory and non-geometric backgrounds is para-Hermitian geometry, an independent branch of differential geometry which has recently become an area of interest because of these applications. This project will explore and develop various features of para-Hermitian geometry with an eye to understanding global aspects of double field theory and non-geometry. Further generalizations of the framework to describe the connections with other topics such as exceptional field theory and non-abelian T-duality are other avenues this project could explore.

Abstract: There has been considerable recent interest in topological phases of matter from both mathematicians and physicists. The 2016 Nobel Prize in Physics was awarded to the theoretical physicists David Thouless, Duncan Haldane, and Michael Kosterlitz whose work established the role of topology in understanding various exotic forms of matter.  Gauge theories with local symmetries appear to provide a mechanism for understanding
some topological phases, including quantum Hall states and quantum spin liquids. This project will investigate theories with so-called subsystem symmetries, intermediate between gauge theories and theories with global symmetries, which are related via a quantum duality to  systems with fracton topological order. Such systems exhibit fractional excitations termed “fractons" which can either be immobile or only free to move in restricted subspaces. The phase structure of the associated classical spin models, typically containing multi-spin interactions,  will also be of interest.

Abstract: Renormalisation group flows describe how quantum field theories (QFTs) change when we change the scale of interaction. The coupling constants then change according to their beta functions that can be considered as a vector field on the space of theories. More geometry on the space of theories arises when one takes into account stress-energy tensor conservation and anomaly consistency conditions. The crucial equation describing the local geometry is a gradient formula for the beta functions. Roughly speaking it expresses the beta function as a gradient of some potential function plus additional terms related to various local tensor fields. In string theory such equations for two-dimensional QFTs can be considered as space-time equations of motion for the string fields. The project includes studying the gradient formulae, local geometry and its string theory interpretation. Possible concrete problems include N=2 supersymmetric and PT-symmetric gradient formula for boundary RG flows in two dimensions. Some references: arXiv:1310.4185, arXiv:0910.3109, arXiv:hep-th/0312197.

Abstract: Integrable quantum  field theories and integrable  quantum spin chains
with  boundaries have  been  a topic  of major  interest  for over  20
years. Applications arise  in both string theory  and condensed matter
physics. The word integrable means that these systems possess enhanced
symmetries - with the consequence that some of their properties can be
computed  exactly.  In particular,  it  is  in principle possible  to
compute their energy eigenvalues  exactly. These eigenvalues are given
in terms of the solution of a system of equations called 'Bethe ansatz
equations',  which in  term come  from  a more  fundamental system  of
equations called  'Baxter relations'. Baxter relations  are difference
equations for a polynomial Q(z). The  modern  construction and  understanding  of  quantum spin chains
relies on  the representation theory  of quantum groups, also know as
quasi-triangular Hopf  algebras. While this picture  is well-developed
for closed,  periodic quantum  spin chains, it  is only  very recently
that Baxter  relations have  been fully  understood in  this language.
The main goal  of this project is to develop  a parallel understanding
of Baxter  relations in 'open'  quantum spin  chains - that  is, those
with two independent integrable boundary conditions.

Abstract: Quantization of locally non-geometric closed string backgrounds reveals a nonassociative deformation of phase space geometry. Nonassociative differential geometry has been developing over the last few years, but there are many aspects still to be understood, particularly its metric aspects. This project concerns the development of nonassociative geometry from various points of view, including quasi-Hopf twist deformation techniques and L-infinity algebras, with an eye to understanding and developing a nonassociative theory of gravity as an effective low-energy theory of closed strings in non-geometric backgrounds.

Abstract: The aim of this PhD project is to device strategies for computing rigorous sharp bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods (Galerkin methods). The theory of computation for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart however a systematic procedure for computing their spectrum remains as a largely unsolved open problem. During this PhD, the student will examine general strategies for numerically estimating spectra of J-self-adjoint operators. Particular directions will involve the following. (a) Computation of enclosures of spectra for operators arising in the theory of rotating viscous fluids. (b) Computation of sharp enclosures for the complex isotonic harmonic oscillator. (c) Impact in the study of time evolution problems for J-self-adjoint operators.

Abstract: Please see the link below for up to date list of projects in mathematical physics at the University of Edinburgh.

Abstract: Renormalisation group (RG) is as fundamental concept in Quantum Field Theory (QFT) that describes how the physics changes under the change of energy scale. A typical renormalisation group trajectory starts from one fixed point and drives the theory to a different fixed point. In two dimensions the fixed points are described by conformal field theories that possess an infinite-dimensional symmetry algebra. While a lot in known about the end points of renormalisation group flows very little is known about the global structure of the space of flows linking the end points. Recently a new object called Renormalisation domain wall or renormalisation defect has been invented. In two dimensions this object is a line of interface between two different conformal field theories on each side. The project involves studying such objects for concrete RG flows both analytically and numerically. The aims are to learn how to construct such objects, what information they encode about RG flows and how could they be used in gaining control over the space of flows. Some references: arXiv:1201.0767; arXiv:1211.3665; arXiv:1407.6444, arXiv:1610.07489.

Abstract: This project is about finding certified bounds for eigenvalues in regimes where the classical methods fail catastrophically. The Galerkin method is among the most reliable tools for finding upper bounds for the eigenvalues of selfadjoint operators. It is widely used in applications, ranging from civil and mechanical engineering to thermodynamics and non-relativistic quantum chemistry. Unfortunately the Galerkin method can fail dramatically when applied to the “wrong” eigenvalue problem. This phenomenon, called spectral pollution, is notoriously difficult to predict and it arises in relativistic quantum mechanics, solid state physics and electromagnetism. The purpose of this PhD project, is to characterise on a mathematically rigorous level what spectral pollution is and then examine analytical and numerical techniques for avoiding it. These include the quadratic method, the ZMG method, the factorisation method and the perturbation method. At the end of the project the student will have a broad view on the state-of-the-art in analytical and computational spectral theory with a specific emphasis in the current topic of spectral pollution.

Abstract: In recent years much progress has been made in understanding of supersymmetric gauge theories. Exact computations have revealed interesting links with quantum integrable systems. In this PhD project we will deepen this connection and employ it to study quantum Hitchin systems.

Abstract: There has recently been rapid progress in experimental techniques for realising gauge fields with interesting topological properties by manipulating  cold atom systems with Laser beams.  It is known how to produce   certain magnetic fields with non-trivial linking numbers, but  there is no general understanding of  the knotted and linked structures which one can produce in practice. The goal of this project is to develop a systematic theory of the laser beams required to produce a given link or knot.

Abstract: Bridgeland stability conditions are a new powerful tool to understand the category of coherent sheaves on varieties. Enough technology now exists for stability conditions on 3-folds of Picard rank 1 to be able to answer questions about the geometry of the underlying variety. For abelian 3-folds we also have a powerful tool in the form of the Fourier-Mukai transform which interacts well with stability conditions. This project would aim to extend the methods used to understand the geometry of abelian surfaces to 3-folds.