The Dirac operator is a first order partial differential operator acting in
the space of vector valued functions $L^2(\mathbb{R}^3; \mathbb{C}^4)$.
It is the relativistic counterpart of the Schr\"odinger operator and
hence, it appears in many applications in quantum mechanics.
In this talk Dirac operators $A_\tau$ with Lorentz scalar $\delta$-shell
interactions of strength $\tau \in \mathbb{R}$ supported on a smooth surface
$\Sigma$ will be discussed. First, self-adjointness and basic spectral properties
of these operators are obtained. Then, it turns out that for this special type of interactions the quadratic form of $A_\tau^2$ can be computed explicitly.
This allows us to find estimates for the discrete eigenvalues of $A_\tau$
for large masses; the eigenvalues are determined by an effective operator on
$\Sigma$ which is a Hamiltonian with an external Yang-Mills potential.
The talk is based on a joint work with T.~Ourmi\`eres-Bonafos and
K.~Pankrashkin.