String Theory

This work falls within the EPSRC mathematical physics research area.

General relativity has enjoyed phenomenological success and underpins modern cosmology. Successes include the prediction of black holes and gravitational radiation and the understanding that our universe's large scale geometry is dynamical.

Despite these considerable results GR is imperfect, in that it has resisted a quantum mechanical description. In four dimensions an attempt to quantise GR in the same manner as electromagnetism's successful description by quantum electrodynamics is met with ultraviolet divergences not treatable by any known methods of renormalisation. Tantalising examples of the interplay between quantum mechanics and GR (the paragon being Hawking radiation) have allowed for several advances, with black hole thermodynamics telling us a lot, but a unification of these two pillars of physics is yet to appear and can certainly be expected to appear vastly different to both. Short distance gravitational interactions should be modified in some way that removes these plaguing divergences while retaining GR at lower energies, and the use of Lagrangian descriptions in quantum field theories is already under question with several quantum theories now known that do not appear to permit these.

The Standard Model accurately describes known physics besides gravity, but does not comment on the origins of its many constants which are treated as fundamental. Any mechanisms behind these values will appear in a fundamental theory, as will the honest inclusion of gravity.

The hope for a consistent quantum field theory incorporating GR is met by string theory. The low energy limits of these theories are supergravity theories that naturally incorporate GR's successes. Additionally, mechanisms have been discovered that allow for Standard Model physics to be included. Few would argue that string theory as it is currently understood is in its final form, with a total description of its degrees of freedom remaining on the horizon. Several important relations remain conjectures- one example being the AdS/CFT correspondence, which claims a relation between two (at first glance) very different kinds of theories. One can expect proofs of this to involve novel methods. This frontier quality of the theory, and the appearances of pure mathematics in the wild of mathematical physics, make this an area rich for projects.

String theory enjoys many relations to other areas of study. Differential geometry's utility is immediate, both on the worldsheet and in the ambient spacetime. Further, the tools of algebraic geometry are indispensable to the string theorist with Calabi Yau manifolds being known to hold great importance when one wishes to relate the ten dimensions that the theory must have to the four dimensions that we know of in a way that retains any supersymmetry. Techniques from number theory have also provided insights- in Moore's work [1-2] number theoretic considerations give a view into the physics of dyonic black holes in the supergravity limit of type IIB string theory. These include using class numbers to count the U-duality distinct charges possible for a given horizon area, describing the BPS spectrum in terms of norms of ideals in quadratic imaginary fields and generalisations of the arithmetic properties of the j-function being realised by the mirror map for K3 surfaces.

Initial project work would relate to a forthcoming paper by Candelas et al., and concerns arithmetic properties of attractor points as a function of a black hole's charges in N=2 supergravity. The immediate project goals are to carry out work in this area, a broad aim being to build on the relationship between number theory and theoretical physics, which is at this point very well established.