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The Donaldson-Thomas invariants are defined by Thomas in the paper A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, following the proposal in Gauge theory in higher dimensions by Donaldson and Thomas. Although the initial proposal was motivated by differential geometry, the rigorous definition uses tools from algebraic geometry, including moduli spaces of (semi-)stable sheaves and perfect obstruction theory.

The most successful application of DT theory so far seems to be the enumeration of curves, i.e. considering the DT invariants of ideal sheaves, see e.g. 13/2 ways of counting curves. So actually the question is two-fold:

  1. Is there a way of counting solutions to some version of (perturbed) Hermitian-Yang-Mills equations which could presumably recover DT invariants of ideal sheaves?

  2. As Gromov-Witten theory also makes sense for symplectic manifolds, is there a symplectic (actually, almost complex) counterpart of DT theory?

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  • $\begingroup$ Theorem 1.4 of Bridgeland-Smith (arxiv.org/abs/1302.7030) implies that the DT invariants of certain quasi-projective CY 3-folds, defined with respect to the stability condition associated to the quadratic differential, are given by counting certain special Lagrangian submanifolds. See allso the work of Joyce: arxiv.org/abs/hep-th/9907013. $\endgroup$
    – YHBKJ
    Commented May 9, 2020 at 23:44

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A symplectic theory should exist but is still some way off, even for curve counting invariants. See arXiv/1712.08383 by Doan-Walpuski for some progress for stable pairs.

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