Donaldson-Thomas invariants for the Bridgeland-Smith correspondence

TL;DR

This paper verifies that the Donaldson-Thomas invariants computed from a specific category align with physics predictions, linking quadratic differential trajectories to stability conditions and invariants.

Contribution

It demonstrates that the Donaldson-Thomas invariants from the Christ-Haiden-Qiu category match physics predictions, using novel string and band techniques.

Findings

Invariants agree with physics predictions.

Degenerate ring domains produce non-zero invariants.

Application of representation theory techniques.

Abstract

Famous work of Bridgeland and Smith shows that certain moduli spaces of quadratic differentials are isomorphic to spaces of stability conditions on particular 3-Calabi-Yau triangulated categories. This result has subsequently been generalised and extended by several authors. One facet of this correspondence is that finite-length trajectories of the quadratic differential are related to categories of semistable objects of the corresponding stability condition, which have associated Donaldson-Thomas invariants. On the other hand, computations in the physics literature suggest certain values of these invariants according to the type of trajectory. In this paper, we show that the category recently constructed by Christ, Haiden, and Qiu gives Donaldson-Thomas invariants which agree with the predictions from physics; in particular, degenerate ring domains of the quadratic differential give rise to non-zero Donaldson-Thomas invariants. In calculating all of the invariants, we obtain a novel application of string and band techniques from representation theory.