Title: Homotopy dimer algebras and cyclic contractions,
Abstract: Dimer algebras arise from a particular type of quiver gauge theory. However,
part of the input to such a theory is the gauge group, and this choice may
impose additional constraints on the algebra. If the gauge group of a dimer
theory is abelian, then the algebra that arises is not actually the dimer
algebra itself, but a particular quotient we introduce called the 'homotopy
algebra'. We show that a homotopy algebra $\Lambda$ on a torus is a dimer
algebra if and only if it is noetherian, and otherwise $\Lambda$ is the
quotient of a dimer algebra by homotopy relations. Stated in physics terms, a
dimer theory is superconformal if and only if the corresponding dimer and
homotopy algebras coincide. We also give an explicit description of the center
of a homotopy algebra in terms of a special subset of its perfect matchings. In
our proofs we introduce formalized notions of Higgsing and the mesonic chiral
ring from quiver gauge theory.