Rank-$N$ Dimer Models on Surfaces

TL;DR

This paper generalizes the web trace theorem for dimer models to higher genus surfaces, expanding the twisted Kasteleyn determinant in terms of webs and incorporating spin structures, with new geometric derivations.

Contribution

It extends the web trace theorem to higher genus surfaces and spin structures, providing a new geometric proof method.

Findings

Generalization of web trace theorem to higher genus surfaces

Expansion of twisted Kasteleyn matrices for spin structures

Alternative geometric derivation of planar web trace theorem

Abstract

The web trace theorem of Douglas, Kenyon, Shi expands the twisted Kasteleyn determinant in terms of traces of webs. We generalize this theorem to higher genus surfaces and expand the twisted Kasteleyn matrices corresponding to spin structures on the surface, analogously to the rank-1 case of Cimasoni, Reshetikhin. In the process of the proof, we give an alternate geometric derivation of the planar web trace theorem, relying on the spin geometry of embedded loops and a `racetrack construction' used to immerse loops in the blowup graph on the surface.