## Abstract

This article investigates a set of partial differential equations, the DT-instanton equations, whose solutions can be regarded as a generalization of the notion of Hermitian–Yang–Mills connections. These equations owe their name to the hope that they may be useful in extending the DT-invariant to the case of symplectic 6-manifolds. In this article, we give the first examples of nonabelian and irreducible DT-instantons on non-Kähler manifolds. These are constructed for all homogeneous almost Hermitian structures on the manifold of full flags in \(\mathbb {C}^3\). Together with the existence result we derive a very explicit classification of homogeneous DT-instantons for such structures. Using this classification we are able to observe phenomena where, by varying the underlying almost Hermitian structure, an irreducible DT-instanton becomes reducible and then disappears. This is a non-Kähler analogue of passing a stability wall, which in string theory can be interpreted as supersymmetry breaking by internal gauge fields as in Anderson et al. (J High Energy Phys 09:026, 2009).

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## Notes

Any nearly Kähler manifold is Einstein with positive scalar curvature. Hence, if it is complete, must actually be compact.

The two irreducible HYM connections existing for \(s<s_0\) are actually gauge equivalent, see Remark 6.2. However, the gauge transformation exchanging them fixed the reducible HYM connection existing at \(s=s_0\).

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## Acknowledgements

We would like to thank Benoit Charbonneau, Gäel Cousin and Lorenzo Foscolo for helpful conversations regarding this article. We are particularly thankful for Benoit Charbonneau’s comments and carefully reading a previous version of this article.

Gonçalo Oliveira is supported by Fundação Serrapilheira 1812-27395, by CNPq grants 428959/2018-0 and 307475/2018-2, and FAPERJ through the program Jovem Cientista do Nosso Estado E-26/202.793/2019.

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## Appendix A. The topology of the bundles \(P_{\beta }\)

### Appendix A. The topology of the bundles \(P_{\beta }\)

Recall that the bundles \(P_{\beta }\) are constructed via \(P_{\beta }= \mathrm{SU}(3) \times _{(T^2, \lambda _{\beta })} \mathrm{SO}(3)\), where

Let \(V_{\beta }=P_{\beta } \times _{SO(3)} \mathbb {R}^3\) be the vector bundle associated with respect to the standard representation *SO*(3)-representation, and consider the \(\mathrm{U}(2)\) bundle

where

and the ismorphism. This has the property that the \(\mathrm{U}(2)\)-adjoint bundle of \(E_{\beta }\) splits as \(\mathfrak {u}_{E_{\beta }} \cong \underline{\mathbb {R}} \oplus V_{\beta }\) and

We shall now compute the Chern classes of the bundles \(E_{\beta }\) using Chern-Weyl theory. For this we must equip \(E_{\beta }\) with a connection which we choose to be the standard invariant connection given by

This has curvature \(F_{\beta } = d\beta \otimes \mathrm {diag}(i,0)\) and so

Furthermore, a computation using the Maurer–Cartan equations shows that

and so in \(H^4(\mathbb {F}_2, \mathbb {Z})\) we have

So, writing \(\beta = k \beta _1 + l \beta _2\) we compute that

while

In particular, when \(\beta \) is one of the roots \(r_1\), \(r_2\), \(r_3\) we respectively obtain

so these three bundles are all topologically different.

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Ball, G., Oliveira, G. The DT-Instanton Equation on Almost Hermitian 6-Manifolds.
*Commun. Math. Phys.* **388**, 819–844 (2021). https://doi.org/10.1007/s00220-021-04206-8

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DOI: https://doi.org/10.1007/s00220-021-04206-8