  We study the moduli space volume of BPS vortices in quiver gauge theories on
compact Riemann surfaces. The existence of BPS vortices imposes constraints on
the quiver gauge theories. We show that the moduli space volume is given by a
vev of a suitable cohomological operator (volume operator) in a supersymmetric
quiver gauge theory, where BPS equations of the vortices are embedded. In the
supersymmetric gauge theory, the moduli space volume is exactly evaluated as a
contour integral by using the localization. Graph theory is useful to construct
the supersymmetric quiver gauge theory and to derive the volume formula. The
contour integral formula of the volume (generalization of the Jeffrey-Kirwan
residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by
the area of the Riemann surface divided by the intrinsic size of the vortex).
We give some examples of various quiver gauge theories and discuss properties
of the moduli space volume in these theories. Our formula are applied to the
volume of the vortex moduli space in the gauged non-linear sigma model with
$CP^N$ target space, which is obtained by a strong coupling limit of a parent
quiver gauge theory. We also discuss a non-Abelian generalization of the quiver
gauge theory and "Abelianization" of the volume formula.
