EPSRC Reference: 
EP/W020939/1 
Title: 
3d N=4 TQFT's 
Principal Investigator: 
Dimofte, Professor T 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematics 
Organisation: 
University of Edinburgh 
Scheme: 
EPSRC Fellowship 
Starts: 
01 September 2022 
Ends: 
31 August 2027 
Value (£): 
881,245

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 
The research in this Fellowship lies at the interface of pure mathematics (algebra geometry and topology) and theoretical physics (quantum field theory). I will construct a new class of threedimensional topological quantum field theories, the eponymous 3d N=4 TQFT's, and use a combination of techniques from physics, algebra, and geometry to understand and define their structure.
A hallmark of a 3d TQFT is that its physical properties only depend on the shape  but not the size  of threedimensional spacetime. A classic example of such a TQFT, called ChernSimons theory, was constructed in the 90's. Its quantum expectation values were used to distinguish shapes of knots and threedimensional spaces.
The 3d N=4 TQFT's I construct come from taking sectors of supersymmetric gauge theories that behave topologically. They are similar to ChernSimons in some ways, but infinitely richer and more complicated in others. On one hand, their spaces of quantum states are infinite rather than finitedimensional, and their expectation values will take a great deal of care to properly define. On the the hand they come in pairs, related by a duality (an equivalence) called 3d Mirror Symmetry, which roughly implies that any one computation can be done in at least two completely different ways, from two different perspectives. In physical terms, 3d Mirror Symmetry says that particles and vortices moving around in three dimensions basically look the same, and will probe the shape of a threedimensional space in equivalent ways.
My research takes such intuitive statements and turns them into rigorous mathematics. It turns out that the mathematical structure of 3d N=4 TQFT's is related to an astounding number of other areas of mathematics  the fields of vertex operator algebras, geometric representation theory, mirror symmetry (an older type, inspired by string theory), and topology all get related in surprising new ways to 3d N=4 TQFT's, to 3d physics, and ultimately to each other.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

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