  We develop a discontinuous cut finite element method (CutFEM) for the
Laplace-Beltrami operator on a hypersurface embedded in $\mathbb{R}^d$. The
method is constructed by using a discontinuous piecewise linear finite element
space defined on a background mesh in $\mathbb{R}^d$. The surface is
approximated by a continuous piecewise linear surface that cuts through the
background mesh in an arbitrary fashion. Then a discontinuous Galerkin method
is formulated on the discrete surface and in order to obtain coercivity,
certain stabilization terms are added on the faces between neighboring elements
that provide control of the discontinuity as well as the jump in the gradient.
We derive optimal a priori error and condition number estimates which are
independent of the positioning of the surface in the background mesh. Finally,
we present numerical examples confirming our theoretical results.
