Dimer quivers on genus g surfaces and noncommutative desingularizations

A dimer algebra is a type of Jacobian algebra whose quiver \(Q\) embeds in a surface \(S\), such that each connected component of \(S\backslash Q\) is simply connected and bounded by an oriented cycle of \(Q\). It was shown in 2009 that noetherian dimer algebras on a torus are noncommutative desingularizations of their centers; in particular, they are ‘homologically smooth’ endomorphism rings. On higher genus surfaces, however, these nice properties disappear. I will introduce special quotients of dimer algebras, called ‘ghor algebras’, where the relations come from the quiver’s perfect matchings rather than a potential. On a torus, a dimer algebra coincides with its ghor algebra if and only if it is noetherian, whereas ghor algebras are almost never noetherian on higher genus surfaces. Nevertheless, I will describe how a ghor algebra, on any genus \(g\) surface, may be viewed as a noncommutative desingularization of its center. This is joint work with Karin Baur.