  Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma$
an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by
quiver invariant theoretic considerations, we introduce the Brascamp-Lieb
operator $T_{V,\sigma}$ associated to $(V,\sigma)$ and study its capacity,
denoted by $\mathbf{D}_Q(V, \sigma)$. When $Q$ is the $m$-subspace quiver, the
capacity of quiver data is intimately related to the Brascamp-Lieb constants
that occur in the $m$-multilinear Brascamp-Lieb inequality in analysis.
  We show that the positivity of $\mathbf{D}_Q(V, \sigma)$ is equivalent to the
$\sigma$-semi-stability of $V$. We also find a character formula for
$\mathbf{D}_Q(V, \sigma)$ whenever it is positive. Our main tool is a quiver
version of a celebrated result of Kempf-Ness on closed orbits in invariant
theory. This result leads us to consider certain real algebraic varieties that
carry information relevant to our main objects of study. It allows us to
express the capacity of quiver data in terms of the character induced by
$\sigma$ and sample points of the varieties involved. Furthermore, we use this
character formula to prove a factorization of the capacity of quiver data. We
also show that the existence of gaussian extremals for $(V, \sigma)$ is
equivalent to $V$ being $\sigma$-polystable, and that the uniqueness of
gaussian extremals implies that $V$ is $\sigma$-stable. Finally, we explain how
to find the gaussian extremals of a gaussian-extremisable datum $(V, \sigma)$
using the algebraic variety associated to $(V,\sigma)$.
