Periodic Reporting for period 3 - StabilityDTCluster (Stability conditions, Donaldson-Thomas invariants and cluster varieties)
Reporting period: 2018-10-01 to 2020-03-31
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.