EPSRC Reference: 
EP/R013349/1 
Title: 
VafaWitten invariants of projective surfaces 
Principal Investigator: 
Thomas, Professor R 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Imperial College London 
Scheme: 
Standard Research 
Starts: 
01 September 2018 
Ends: 
31 August 2021 
Value (£): 
707,021

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 
In 1994 the physicists Vafa and Witten introduced new "invariants" of four dimensional spaces. These invariants "count" solutions of a certain equation (the N=4 supersymmetric YangMills equations) over the four dimensional space, and should tell us something about the space. There is one for every integer charge of the YangMills field.
Motivated by a generalisation of electromagnetic duality in string theory, Vafa and Witten predicted that on a fixed space, one could put all these invariants together in a generating series (a Taylor series or Fourier series, with coefficients the VafaWitten invariants) and get a very special function called a "modular form". In particular the invariants should have hidden symmetries that mean that only a finite number of them determine all the rest.
Until now mathematicians have been unable to make sense of how this "counting" should be done without getting infinity. This project gives a definition for any space which is "projective", and for any charge (including ones for which troublesome "semistable" or "reducible" solutions appear). We will then compute the invariants for many such spaces with negative curvature. We will also produce "refined" VafaWitten invariants containing more information. These should be the invariants sought by physicists aiming to describe "topologically twisted maximally supersymmetric 5d super YangMills theory".

Key Findings 
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Impacts 
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Summary 

Date Materialised 


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Further Information: 

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