Riemann–Hilbert problems for the resolved conifold and non-perturbative partition functions

We study the Riemann-Hilbert problems of [6] (T. Bridgeland, “Riemann-Hilbert problems from Donaldson–Thomas theory”, arxiv:1611.03697) in the case of the Donaldson–Thomas theory of the resolved conifold. We give explicit solutions in terms of the Barnes double and triple sine functions. We show that the τ-function of [6] is a non-perturbative partition function, in the sense that its asymptotic expansion coincides with the topological closed string partition function

Scattering diagrams, Hall algebras and stability conditions

With any quiver with relations, we associate a consistent scattering diagram taking values in the motivic Hall algebra of its category of representations. We show that the chamber structure of this scattering diagram coincides with the natural chamber struc- ture in an open subset of the space of stability conditions on the associated triangulated category. In the three-dimensional Calabi–Yau situation, when the relations arise from a potential, we can apply an integration map to give a consistent scattering diagram taking values in a tropical vertex group

Hall algebras and Donaldson-Thomas invariants

This is a survey article on Hall algebras and their applications to the study of motivic invariants of moduli spaces of coherent sheaves on Calabi-Yau threefolds. It is a write-up of my talks at the 2015 Salt Lake City AMS Summer Research Institute and will appear in the Proceedings. The ideas presented here are mostly due to Joyce, Kontsevich, Reineke, Soibelman and Toda

Geometry from Donaldson-Thomas invariants

We introduce geometric structures on the space of stability conditions of a three-dimensional Calabi-Yau category which encode the Donaldson-Thomas invariants of the category. We explain in detail a close analogy between these structures, which we call Joyce structures, and Frobenius structures. In the second half of the paper we give explicit calculations of Joyce structures in three interesting classes of examples

The monodromy of meromorphic projective structures

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on the associated marked bordered surface. We prove that the image of this map is contained in the union of the domains of the cluster charts. We discuss a number of open questions concerning this monodromy map

Derived automorphism groups of K3 surfaces of Picard rank 1

We give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a K3 surface of Picard rank 1. We do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences, and is contractible

Complex hyperkähler structures defined by Donaldson–Thomas invariants

The notion of a Joyce structure was introduced in Bridgeland (Geometry from Donaldson–Thomas invariants, preprint arXiv:1912.06504) to describe the geometric structure on the space of stability conditions of a CY3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms

Stability conditions and the A2 quiver

For each integer n≥2 we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the singularity

On the monodromy of the deformed cubic oscillator

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Riemann-Hilbert problems of (Bridgeland in Invent Math 216(1):69–124, 2019) arising from the Donaldson-Thomas theory of the A2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map