We study a class of flat bundles, of finite rank $N$, which arise naturally from the Donaldson–Thomas theory of a Calabi–Yau threefold $X$ via the notion of a variation of BPS structure. We prove that in a large $N$ limit their flat sections converge to the solutions to certain infinite-dimensional Riemann–Hilbert problems recently found by Bridgeland. In particular this implies an expression for the positive degree, genus 0 Gopakumar–Vafa contribution to the Gromov–Witten partition function of $X$ in terms of solutions to confluent hypergeometric differential equations.

We study the action of the inertia operator on the motivic Hall algebra and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.

We study the relative Donaldson–Thomas theory of 𝒜n×P1, where 𝒜n is the surface resolution of type An singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov–Witten theory of 𝒜n×P1 and the quantum cohomology of the Hilbert scheme of points on 𝒜n.