A dimer primer

I’m now mostly settled in Japan – so far in my downtime I’ve learned hiragana and am up to タ in katakana, but shopping can still be rather challenging.. – though I haven’t been working too seriously before a week-long conference in Kyoto starting on Monday. Here’s my poster for it if you’re interested.

I went to a seminar talk by Alvaro Nolla de Celis this week on dimer models in geometry. Much like seeing Michael Wemyss speak (that I finally managed at BrAG), managing to see a dimer talk has been eluding me for a while; there’s always been some schedule clash meaning that I couldn’t go. I defied my usual practice by taking notes and so used them to base this post.

It seems that using moduli spaces of quiver representations with some stability condition to resolve singularities has been gaining momentum in algebraic geometry for a while. One of the problems with this method is that it is not always clear how to associate a quiver to a singularity. In the McKay correspondence there is a natural choice – the McKay quiver of the group giving the quotient singularity – and one can view all of this as a generalisation of that procedure.

Dimer models are bipartite graphs drawn on an oriented 2-torus. That’s a compact, real 2-torus rather than an algebraic torus.. Lifting them to the universal cover of the torus produces a tiling of the plane by (possibly different) polygons. The first piece of data comes from ‘zigzag paths’ through the dimer. These are 1-cycles defining classes in H_1(T^2)\cong\mathbb{Z}^2. By collecting them all together, one obtains some list of lattice points in the plane. This list is quite special though: there are exactly two zigzag paths through any edge of the dimer – one in each direction – and so the sum of these classes is zero. That means that there’s a polygon for which the outward normals to its sides are the listed vectors. This is the ‘zigzag polygon’. The second piece of data is what is essentially the oriented dual graph, which is a quiver. For a dimer \Gamma, let P_\Gamma be its zigzag polygon and Q_\Gamma be its quiver. There are a slew of interesting results in commutative and noncommutative geometry going between these two objects in various guises, but the one I focused on is due to Ishii-Ueda…

Theorem: The moduli space of \theta-semistable representations of Q_\Gamma is a crepant (or Calabi-Yau) resolution of the toric singularity X_{\mathcal{C}P_\Gamma} for generic \theta.

One constructs the singularity – which is actually affine Gorenstein – by inserting P_\Gamma at height 1 in \mathbb{R}^3 and taking the cone over it to get a polyhedral cone, then consider the corresponding affine toric variety. If you can’t see why this is Gorenstein, go back to my notes from a few months ago!

This theorem incorporates some versions of the McKay correspondence when P_\Gamma is a triangle and Q_\Gamma is the McKay quiver as discussed. Interestingly, it was also proved without appealing to the original results of Bridgeland-King-Reid. Anyway, I hope that you have as much fun as I did toying with dimers.

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