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Stability conditions, Donaldson-Thomas invariants and cluster varieties

Periodic Reporting for period 3 - StabilityDTCluster (Stability conditions, Donaldson-Thomas invariants and cluster varieties)

Reporting period: 2018-10-01 to 2020-03-31

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.
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) foundational work in Donaldson-Thomas theory. We have made contact with several other disciplines of pure mathematics, including the geometry of Riemann surfaces, as well as proving results of direct relevance in string theory. Further details can be found in the scientific report.

Two results are of particular importance. Firstly, 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.

Secondly, in string theory, certain numbers called BPS invariants play an important role. They are indices of certain operators, and hence by definition, integers. In very important work around ten years ago, mathematicians managed to come up with a rigorous definition of these numbers using Donaldson-Thomas theory. There was a catch however: the numbers mathematicians defined could in principle be rational numbers (fractions), and it became an important open problem to prove that they were in fact integers. One of the RAs on the grant (Sven Meinhardt) working with a collaborator Ben Davison has now proved this conjecture.
The link with non-perturbative expansions in string theory was unexpected and we are working hard to try to generalise this result. We are also looking at quantum versions of the Riemann-Hilbert problem. The work relating to the geometry of Riemann surfaces is continuing, and Meinhardt is continuing his foundational studies of Donaldson-Thomas theory. Again, further details can be found in the scientific report.