  We investigate ghost-free vector-tensor theories in metric-affine geometry.
In all of our analysis, we start with the Lagrangian containing up to quadratic
terms of first-order derivatives of a vector field. To obtain ghost-free
vector-tensor theories efficiently, we consider two options; the theories
satisfy the torsionless condition or have the projective symmetry. We first
explore the vector-tensor theories under the former condition. We then
investigate the projective-invariant vector-tensor theories in metric-affine
geometry. To systematically construct a projective-invariant Lagrangian, we use
two different approaches. First, we construct a Lagrangian by contracting the
epsilon tensor. Second, we construct a Lagrangian by use of
projective-invariant combinations. We find that to obtain a ghost-free
Lagrangian in metric-affine geometry, imposing the projective invariance would
be more useful than imposing the torsionless condition. However, we also prove
that the projective invariance (or the torsionless condition) alone is
insufficient for vector-tensor theories in metric-affine geometry to be
ghost-free.
