Examples of deformed \(\operatorname{Spin}(7)\)-instantons/Donaldson-Thomas connections (Q6585684)

Given a Kähler manifold \((M^{2n}, g, J, \omega)\) and a Hermitian line bundle \(L \to M\), a connection \(1\)-form \(A\) on \(L\) is called a deformed Hermitian Yang-Mills (dHYM) connection with phase \(e^{i\theta}\) if its curvature \(F_A\), viewed as a real 2-form, satisfies \[F_A^{0,2} = 0, \Im((\omega + i F_A )^n) = \tan(\theta)\Re((\omega + i F_A)^n )\] for some constant \(\theta\) (see [\textit{N. C. Leung} et al., Adv. Theor. Math. Phys. 4, No. 6, 1319--1341 (2000; Zbl 1033.53044); \textit{M. Mariño} et al., J. High Energy Phys. 2000, No. 1, Paper No. 01(2000)005, 31 p. (2000; Zbl 0990.81585)])\N\NThe author states that dHYM connections have received considerable interest in recent years and now many examples on compact Kähler manifolds are known (see e.g., [\textit{G. Chen}, Invent. Math. 225, No. 2, 529--602 (2021; Zbl 1481.53093)] but ``by contrast, very little is known in the context of \(\mathrm{G}_2\) and \(\mathrm{Spin}(7)\) geometry''. From the abstract: ``We construct examples of deformed Hermitian Yang-Mills connections and deformed \(\mathrm{Spin}(7)\)-instantons (also called \(\mathrm{Spin}(7)\) deformed Donaldson-Thomas connections) on the cotangent bundle of \(\mathbb{C}P^2\) endowed with the Calabi hyperKähler structure. Deformed \(\mathrm{Spin}(7)\)-instantons on cones over \(3\)-Sasakian \(7\)-manifolds are also constructed. We show that these can be used to distinguish between isometric structures and also between \(\mathrm{Sp}(2)\) and \(\mathrm{Spin}(7)\) holonomy cones. To the best of our knowledge, these are the first non-trivial examples of deformed \(\mathrm{Spin}(7)\)-instantons.''