EPSRC Reference: 
EP/L001527/1 
Title: 
Singular spaces of special and exceptional holonomy. 
Principal Investigator: 
Haskins, Professor M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Dept of Mathematics 
Organisation: 
Imperial College London 
Scheme: 
Standard Research 
Starts: 
01 July 2013 
Ends: 
30 September 2015 
Value (£): 
253,271

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
22 May 2013

Developing Leaders Meeting  LF

Announced


Summary on Grant Application Form 
M theory is an 11dimensional theory in theoretical physics; it is thought a promising candidate for a consistent quantum theory of gravity, i.e. a theory that unites quantum mechanics with Einstein's theory of General Relativity in a consistent way. To obtain 4dimensional universes like ours from an 11dimensional theory one postulates that the ``extra 7 dimensions'' are small. This process is called compactification and the small 7dimensional space is called the compactifying space. At low enough energy scales the physics is then 4dimensional and the properties of the 4dimensional physics is determined by the features of the compactifying space. M theory compactifications on special 7dimensional spacescalled manifolds with G2holonomyattracted particular attention because of their potential to incorporate theories that can accurately describe all the currently known fundamental particles in nature into a unified theory containing gravity.
These same G2 holonomy spaces had already been studied for many years by mathematicians studying geometry. Geometers knew that if they could find such spaces then they would have very special geometric properties involving their curvature. Geometers call these spaces Ricciflat because some part of their curvature, their socalled Ricci curvature vanishes. Ricci curvature also plays an important role in the basic equations in General Relativity. For this reason mathematicians regard Ricciflat spaces as very special; they are like analogues of the spaces in General Relativity one would see if no matter were present. However Ricciflat spaces that were not already totally flat proved very hard to find and none were known until 1978. Manifolds with G2 holonomy (also called G2manifolds) proved even harder to find; it wasn't until the mid 1990s that Joyce found a way to produce G2manifolds of finite extent. Finding such G2manifolds was considered a major achievement and involved solving systems of socalled nonlinear partial differential equations in a rather indirect wayby first solving different and very slightly easier equations, and then proving that an appropriate small adjustment of this solution would solve the original system of equations.
However, when theoretical physicists started studying the physical properties of M theory compactifications on the G2manifolds found by Joyce, they realised there was a problem. The physical theories they got out turned out not to be compatible with the basic known facts about the fundamental particles. Later other theoretical physicists reexamined these problems in M theory and realised a way out of their dilemma. If the 7dimensional compactifying space still had the special Riccicurvature property described, but in addition had some very special points that look different from surrounding points and at which the full curvature can be infinite, then the physicists could get more complicated and realistic theories. Geometers called these special points, singularities, because of the curvature being infinite there. If M theorists supposed that singular G2 holonomy spaces with very special kinds of singularities existed, then they found that they got out physics compatible with the basic known facts about the fundamental particles.
The only problem was that mathematicians could no longer demonstrate that such singular G2 holonomy spaces exist. The method Joyce had pioneered broke down in the presence of the singularities that the physicists needed to get out realistic physics. Even today geometers have still not be able to demonstrate that the singular G2 spaces needed by M theorists exist. This proposal sets out to develop the new mathematics needed to find these kinds of singular G2 spaces (and other singular spaces with similar curvature properties) as part of a collaborative project involving both geometers and M theorists.

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Organisation Website: 
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