We propose a generalization of the Bogomol’ny–Prasad–Sommerfield Skyrme model [L. A. Ferreira, Exact self-duality in a modified Skyrme model, J. High Energy Phys. 07 (2017) 039] for simple compact Lie groups $G$ that leads to Hermitian symmetric spaces. In such a theory, the Skyrme field takes its values in $G$, while the remaining fields correspond to the entries of a symmetric, positive, and invertible $\dim G\times \dim G$-dimensional matrix $h$. We also use the holomorphic map Ansatz between $S^2\to G/H\otimes U(1)$ proposed in Ferreira and Livramento [Harmonic, holomorphic and rational maps from self-duality, arXiv:2412.02636] to study the self-dual sector of the theory, which generalizes the holomorphic Ansatz between $S^2\to C{P}^N$ proposed in Ioannidou [Low-energy states in the SU(N) Skyrme models, in International Meeting on Mathematical Methods in Modern Theoretical Physics (ISPM 98) (1998), pp. 91–123, arXiv:hep-th/9811071]. This Ansatz is constructed using the fact that stable harmonic maps of the two $S^2$ spheres for compact Hermitian symmetric spaces are holomorphic or antiholomorphic [J. Eells and L. Lemaire, Two Reports on Harmonic Maps (World Scientific Publishing Company, Singapore, 1995)]. Apart from some special cases, the self-duality equations do not fix the matrix $h$ entirely in terms of the Skyrme field, which is completely free, as it happens in the original self-dual Skyrme model for $G=SU(2)$. In general, the freedom of the $h$ fields tend to grow with the dimension of $G$. The holomorphic Ansatz enable us to construct an infinite number of exact self-dual Skyrmions for each integer value of the topological charge and for each value of $N\ge 1$, in case of the $C{P}^N$, and for each values of $p$, $q\ge 1$ in case of $SU(p+q)/SU(p)\otimes SU(q)\otimes U(1)$.

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• Classical solutions in field theory
• Conformal field theory
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• Topological field theories