All the magnetically charged ultrastatic and spherically symmetric spacetime solutions in the framework of linear/nonlinear electrodynamics, with an arbitrary electromagnetic Lagrangian density $\mathcal{ L}(\mathcal{ F})$ depending only of the electromagnetic invariant $\mathcal{ F}\!=\!F_{ \alpha\beta}F^{ \alpha\beta}\!/4$, minimally coupled to Einstein-scalar-Gauss-Bonnet gravity [EsGB-$\mathcal{ L}(\mathcal{ F})$], are found. We also show that a magnetically charged ultrastatic and spherically symmetric EsGB-$\mathcal{ L}(\mathcal{ F})$ solution with invariant $\mathcal{ F}$ having a strict global maximum value $\mathcal{ F}_{ _{ 0}}$ in the entire domain of the solution, and such that $\mathcal{ L}_{ _{ 0}}=\mathcal{ L}(\mathcal{ F}_{ _{ 0}})>0$, can be interpreted as an ultrastatic wormhole spacetime geometry with throat radius determined by the scalar charge and the quantity $\mathcal{ L}_{ _{ 0}}$. We provide some examples, including Maxwell's theory of electrodynamics (linear electrodynamics) $\mathcal{ L}_{ _{ _{ \mathrm{ LED}}}} \!=\! \mathcal{ F}$, producing the magnetic dual of the purely electric Ellis-Bronnikov EsGB Maxwell wormhole derived in [P. Ca\~nate, J. Sultana, D. Kazanas, Phys. Rev. D { \bf100}, 064007 (2019)]; and the nonlinear electrodynamics (NLED) models given by Born-Infeld $\mathcal{ L}_{ _{ _{ \mathrm{ BI}}}} \!=\! -4\beta^{ 2} + 4\beta^{ 2} \sqrt{ 1 + \mathcal{ F}\!/\!(2\beta^{ 2})~}$, and Euler-Heisenberg in the approximation of the weak-field limit $\mathcal{ L}_{ _{ _{ \mathrm{ EH}}}} \!=\! \mathcal{ L}_{ _{ _{ \mathrm{ LED}}}} + \gamma \mathcal{ F}^{ 2}\!/2$. With those NLED models, two novel magnetically charged ultrastatic traversable wormholes (EsGB Born-Infeld and EsGB Euler-Heisenberg wormholes) are presented as exact solutions without exotic matter in EsGB-$\mathcal{ L}(\mathcal{ F})$ gravity.
Categories: gr-qc