DonaldsonThomas invariants for the BridgelandSmith correspondence
Abstract
Famous work of Bridgeland and Smith shows that certain moduli spaces of quadratic differentials are isomorphic to spaces of stability conditions on particular 3CalabiYau triangulated categories. This result has subsequently been generalised and extended by several authors. One facet of this correspondence is that finitelength trajectories of the quadratic differential are related to categories of semistable objects of the corresponding stability condition, which have associated DonaldsonThomas 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 DonaldsonThomas invariants which agree with the predictions from physics; in particular, degenerate ring domains of the quadratic differential give rise to nonzero DonaldsonThomas invariants. In calculating all of the invariants, we obtain a novel application of string and band techniques from representation theory.
 Publication:

arXiv eprints
 Pub Date:
 January 2024
 DOI:
 10.48550/arXiv.2401.10093
 arXiv:
 arXiv:2401.10093
 Bibcode:
 2024arXiv240110093K
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematics  Geometric Topology;
 Mathematics  Representation Theory;
 14D20;
 14N35;
 18E30;
 57M50;
 81T20
 EPrint:
 54 pages, 13 figures, 1 table