Pure & Applied Mathematics
http://hdl.handle.net/2262/59
Pure & Applied Mathematics2022-08-18T00:55:42ZCharacters, coadjoint orbits and Duistermaat-Heckman integrals
http://hdl.handle.net/2262/98611
Characters, coadjoint orbits and Duistermaat-Heckman integrals
Shatashvili, Samson
The asymptotics of characters of irreducible representations of a compact Lie group G for large values of the scaling factor k are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of G. This phenomenon generalises to coadjoint orbits of central extensions of loop groups and of diffeomorphisms of the circle
. We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contribution which can be interpreted as a formal DH orbital integral.
For some Virasoro modules, our results match formal DH integrals recently computed by Stanford and Witten. In this case, the k-scaling has the same origin as the one which gives rise to classical conformal blocks. Furthermore, we consider reduced spaces of Virasoro coadjoint orbits and we suggest a new invariant which replaces symplectic volume in the infinite dimensional situation. We also consider other modules of the Virasoro algebra (in particular, the modules corresponding to minimal models) and we obtain DH-type expressions which do not correspond to any Virasoro coadjoint orbits.
We introduce volume functions
corresponding to formal DH integrals over coadjoint orbits of the Virasoro algebra and show that they are related by the Hankel transform to spectral densities recently studied by Saad, Shenker and Stanford.
2021-01-01T00:00:00ZJordan systems, bounded symmetric domains and associated group orbits with holomorphic and CR extension theory
http://hdl.handle.net/2262/98494
Jordan systems, bounded symmetric domains and associated group orbits with holomorphic and CR extension theory
Matthews, John Alphonsus
The first chapter will deal with the one to one correspondence between the positive hermitian Jordan triple systems and the bounded symmetric domains. We start by defining the various Jordan systems. Then we continue by giving classification results leading to the classification of the Jordan pairs under finiteness conditions. This is followed by defining bounded symmetric domains, and some motivation on why they became a topic of interest. Up to this point we will be dealing with the Jordan systems over a general ring of scalars, we now restrict ourselves to the complex number field. We show bounded symmetric domains are connected to the positive hermitian Jordan triple systems. A look at Peirce decompositions of the Jordan systems will prove important. With this completed, we deal with the one to one correspondence mentioned. This will involve some results relating to Riemannian manifolds and Lie theory, we can then present the result as presented by Ottmar Loos in [Loos1]. A table of the classification is also included to conclude the chapter. We note here that throughout this thesis when we say we are dealing with bounded symmetric domains we mean circled bounded symmetric domains. In the second chapter we look at some holomorphic extension theory. This is started by looking at and defining manifolds and varieties. We then look at normal spaces and normalisations of spaces with some extension theory. After that we look briefly at Stein manifolds and domains of holomorphy, explaining their importance in relation to (maximal) holomorphic and CR extension theory. We will also look at a CR extension theorem due to Albert Boggess and John Polking. To complete this section we will present a application to the extension theorem from the paper [KaupZait] by Dimitri Zaitsev and Wilhelm Kaup. Our third chapter is based on [KaupZait] giving an overview of results contained within. We will look at orbits associated with bounded symmetric domains with respect to (the connected identity component of) the automorphism group of the domain. We look at various hulls and domains that are relevant to the extension theory results of the paper. Then we state the results, and sketch the extension theorem proof. In the final chapter we will generalise these results. In [KaupZait] , the results are for irreducible bounded symmetric domains, we will look at what happens in the reducible case. We will introduce a generalised notation and generalise the results from the irreducible case when needed.
2006-01-01T00:00:00ZCoherent states and classical radiative observables in the S-matrix formalism
http://hdl.handle.net/2262/98491
Coherent states and classical radiative observables in the S-matrix formalism
Gonzo, Riccardo
In this thesis, we study classical radiative observables perturbatively in terms of
on-shell scattering amplitudes. In particular, we focus primarily on the two-body
problem in gauge and gravitational theories by using an effective field theory ap-
proach. The Kosower-Maybee-O Connell (KMOC) approach, which follows from the
classical on-shell reduction of the in-in formalism by using appropriate massive particle
wavefunctions, is extended to include classical waves which are naturally described by
coherent states. Global observables like the impulse and localized observables like the
waveform and gravitational event shapes are then studied in the amplitude approach,
making contact also with asymptotic symmetries and light-ray operators defined near
null infinity. The classical factorization of radiative observables from the uncertainty
principle is proved to be equivalent to a Poisson distribution in the Fock space, and
this provides new evidence in favor of a representation of the classical S-matrix in
terms of an eikonal phase and a coherent state of gravitons.
APPROVED
2022-01-01T00:00:00ZA performance study of a template C++ class for parallel Monte Carlo simulations of local statistical field theories on a three dimensional lattice
http://hdl.handle.net/2262/98428
A performance study of a template C++ class for parallel Monte Carlo simulations of local statistical field theories on a three dimensional lattice
Burke, Liam
In this thesis we investigate the performance properties of a template C++ class
designed to run parallel Monte Carlo simulations of local statistical field theories on
a three dimensional lattice. The generic nature of the class allows for data type flexi-
bility when defining the mater fields at every site of the grid while incorporating this
flexibility into a generic MPI exchange function to allow for correct data transfer in
parallel simulations. This allows one to overcome the book-keeping issues associated
with the parallel software development of code needed to run simulations of different
field theories with different physical properties. We will investigate how the system
performs in parallel by looking at its scaling behaviour for different matter fields and
will examine the factors affecting its performance - particularly in relation to how
the lattice is stored in memory and how this could influence an optimal choice of
MPI decomposition. To test the performance of our class, we will run parallel sim-
ulations of the 3D Ising model and investigate its critical behaviour by computing
observables such as average magnetization per spin, magnetic susceptibility and its
average energy. The case of a double precision data type on the lattice sites will
then be tested by running simulations of φ4 quantum field theory in 2 + 1 dimensions
using its discrete lattice action. We will run simulations in both the Ising limit and
the limit of a free scalar field theory, and examine how the mass of a particle behaves
as the action parameters are varied. We compare our results to that obtained in the
literature and comment on the similarities to our Ising model results.
2020-01-01T00:00:00Z