210 research outputs found

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

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    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

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    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

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    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

    The monodromy of meromorphic projective structures

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    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 PGL2(C)\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

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    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 hyperkĂ€hler structures defined by Donaldson–Thomas invariants

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    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 hyperkĂ€hler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkĂ€hler metrics in geometric terms

    On the monodromy of the deformed cubic oscillator

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    We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first PainlevĂ© 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

    Stability conditions and the A2 quiver

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    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