By Nathan Broomhead
Quantity 215, quantity 1011 (second of five numbers).
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Extra info for Dimer models and Calabi-Yau algebras
The 2-torus is the quotient of the plane by the fundamental group π1 (T ) which is isomorphic to H1 (T ) as it is abelian. The action is by deck transformations. Given a point x on the plane and a homology class λ we ﬁnd a curve on the torus with this homology class which passes through the projection of x. x to be the end point. This depends only on the homology class, and not on the choice of curve. We note that in particular, the action of [η] ∈ H1 (Q) on an arrow ηn in is the length of one a representative zig-zag ﬂow η, is the arrow ηn+ , where period of η.
If they did, then there is an arrow a ∈ Q which is (without loss of generality) a zig of η and a zag of η . This projects down to an arrow which is both a zig and a zag of η. Thus η intersects itself and the corresponding train track is not a simple closed curve. This contradicts (1), and so η and η do not intersect. This proves part of condition (c). Now we prove that (a) holds. A zig-zag ﬂow can intersect itself in two ways. If an arrow a occurs as a zig and a zag in η, then η projects down to a zig-zag path η which contains the image of a as a zig and a zag.
First we prove that P (σ) evaluates to zero on the two systems of boundary paths. Let η be a representative zig-zag path of γ + or γ − . Since P (σ) is a perfect matching we know that it is non-zero on precisely one arrow in the boundary of every face. By construction, the boundary path of a zig-zag path was pieced together from paths back around faces which have a zig-zag or zag-zig pair in the boundary. 23, P (σ) is non-zero on either the zig or zag of η in the boundary of each of these faces and therefore it is zero on all the arrows in the boundary path.
Dimer models and Calabi-Yau algebras by Nathan Broomhead