Nonlocality and the central geometry of dimer algebras
Abstract
Let $A$ be a dimer algebra and $Z$ its center. It is well known that if $A$ is cancellative, then $A$ and $Z$ are noetherian and $A$ is a finitely generated $Z$module. Here we show the converse: if $A$ is noncancellative (as almost all dimer algebras are), then $A$ and $Z$ are nonnoetherian and $A$ is an infinitely generated $Z$module. Although $Z$ is nonnoetherian, we show that it nonetheless has Krull dimension 3 and is generically noetherian. Furthermore, we show that the reduced center is the coordinate ring for a Gorenstein algebraic variety with the strange property that it contains precisely one 'smearedout' point of positive geometric dimension. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.1750
 Bibcode:
 2014arXiv1412.1750B
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematics  Commutative Algebra;
 Mathematics  Rings and Algebras
 EPrint:
 75 pages. Two new sections and an appendix (originally in 1301.7059v1) have been added