  A ghor algebra is the path algebra of a dimer quiver on a surface, modulo
relations that come from the perfect matchings of its quiver. Such algebras
arise from abelian quiver gauge theories in physics. We show that a ghor
algebra $\Lambda$ on a torus is a dimer algebra (a quiver with potential) if
and only if it is noetherian, and otherwise $\Lambda$ is the quotient of a
dimer algebra by homotopy relations. Furthermore, we classify the simple
$\Lambda$-modules of maximal dimension and give an explicit description of the
center of $\Lambda$ using a special subset of perfect matchings. In our proofs
we introduce formalized notions of Higgsing and the mesonic chiral ring from
quiver gauge theory.
