EPSRC Reference: 
EP/D077990/1 
Title: 
Generalized DonaldsonThomas invariants 
Principal Investigator: 
Joyce, Professor D 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Institute 
Organisation: 
University of Oxford 
Scheme: 
Standard Research 
Starts: 
27 October 2006 
Ends: 
26 April 2010 
Value (£): 
320,460

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
CalabiYau 3folds are a special kind of 6dimensional curved space, with a lot of geometrical structure. They are of great interest to mathematicians working in Algebraic and Differential Geometry, and to physicists working in String Theory. The greatest problem in fundamental physics is to find a single theory which successfully combines Einstein's General Relativity  the physics of very large things, such as galaxies  and Quantum Theory  the physics of very small things, such as atoms. String Theory is the leading candidate for doing this. It predicts that the dimension of spacetime is not 4 (3 space plus one time), but 10. The extra 6 dimensions are rolled up in a CalabiYau 3fold, with very small radius. So according to String Theory, CalabiYau 3folds describe the vacuum of space itself. Using physical reasoning, String Theorists made extraordinary mathematical predictions about CalabiYau 3folds, known as Mirror Symmetry , which have been verified in many cases, and cause much excitement among mathematicians. DonaldsonThomas invariants are systems of numbers associated to a CalabiYau 3fold M which count some mathematical objects ( semistable coherent sheaves ) which live on M. The definition is complicated. They are mathematically interesting because they are unchanged under continuous deformations of M, and encode mysterious, nontrivial information about M. They are physically interesting as they count physically important objects (branes, BPS states). It is at present only known how to define DonaldsonThomas invariants in a special case (when semistable and stable coincide). We propose to find out how to extend the definition to the general case. We also aim to find the transformation laws for these extended invariants under change of stability condition (this is not known even for the old invariants), and to compute them in examples. We hope this will lead to a better understanding of the space of stability conditions, which is part of the space of String Theory vacua, a very important but poorly understood space.

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Organisation Website: 
http://www.ox.ac.uk 