Stability conditions, Donaldson-Thomas invariants and cluster varieties

Project Information

StabilityDTCluster

Grant agreement ID: 670298

DOI

10.3030/670298

Project closed

EC signature date 21 September 2015

Start date 1 October 2015

End date 30 September 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 556 550,00

EU contribution

€ 1 556 550,00

Coordinated by

THE UNIVERSITY OF SHEFFIELD
United Kingdom

Periodic Reporting for period 4 - StabilityDTCluster (Stability conditions, Donaldson-Thomas invariants and cluster varieties)

Reporting period: 2020-04-01 to 2021-09-30

Summary of the context and overall objectives of the project

This is a project in pure mathematics, taking its inspiration from structures in theoretical physics. Quantum field theories have been studied by theoretical physicists for over half a century and have revolutionised the subject. Predictions of quantum field theory have also had a major effect on many areas of pure mathematics over the last twenty years. Unfortunately there is currently no conceptual mathematical understanding of quantum field theory, and the precise calculations which theoretical physicists are able to do have no rigorous basis in pure mathematics. The general aim of this project is to increase our understanding of the mathematics behind quantum field theory. We intend to do this by studying the mysterious relationships between three rigorously defined mathematical objects, each of which was inspired by constructions from theoretical physics. These objects are called stability conditions on triangulated categories, cluster varieties, and Donaldson-Thomas invariants. The key idea to relate them, is something called a Riemann-Hilbert problem.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

There have been three main strands of progress: (i) the abstract study of the Riemann-Hilbert problem; (ii) the geometric study of the stability to cluster map for triangulated surfaces; (iii) links with hyperkahler geometry. We have made contact with several other disciplines of pure mathematics, as well as proving results of direct relevance in string theory. Further details can be found in the scientific report.

One result is of particular importance. In quantum field theory it is common to expand quantities in power series - infinite sums of terms. One of the problems in the subject is that these series are typically divergent - they cannot be directly summed to give finite predictions. We have discovered that in the context of topological string theory, using the Riemann-Hilbert problem, we can write down genuine analytically varying quantities, which somehow `sum' the divergent series, in that their asymptotic expansion is the given power series. Theoretical physicists have looked for such `non-perturbative partition functions' but our approach seems better-motivated and more precise.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The link with non-perturbative partition functions in string theory was unexpected and has lead to a fair amount of interest in the theoretical physics community. The connection with hyperkahler geometry is also an exciting discovery. Work on both of these topics is ongoing, and has attracted further funding.

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