Mirror symmetry is a prediction arising from theoretical physics which roughly conjectures that if we are given a pair of mirror Calabi-Yau varieties, called mirror partners, then the symplectic geometry of one of them (the Fukaya category) is somehow ``reflected'' on the holomorphic/algebraic geometry of the other (derived category of coherent sheaves). It is usually hard to find such mirror partners, but one important breakthrough was proposed by Strominger, Yau and Zaslow (SYZ) in the 90's, who established conditions to construct them.

It turns out that the moduli spaces of Higgs bundles, for Langlands dual Lie groups, verify the SYZ conditions, being now one of the major examples of mirror partners. Moreover, thanks to their hyperkähler structure, mirror symmetry can be realized in terms of tools from algebraic geometry. In this talk we shall consider special subvarieties of these moduli spaces, called branes. These are classified into four types, two of them being $BBB$-branes and $BAA$-branes. Around 10 years ago, Kapustin and Witten conjectured that mirror symmetry should exchange these two types of branes, and this has been verified in some cases, however only in a partial way.

We will introduce these objects and briefly describe a collection of $BBB$- and $BAA$-branes on the moduli of Higgs bundles (for the self-dual group $\mathrm{GL}(n,\mathbb C)$) and how we proved, in a more complete way, that indeed mirror symmetry exchanges them. This is joint work with E. Franco, P. Gothen and A. Peón-Nieto.