Planar dimers and Harnack curves.

*(English)*Zbl 1100.14047Let \(\Gamma\) be the hexagonal lattice of \(\mathbb R^2\) with \(d\times d\)-fundamental domain whose edges are weighted (with real positive weights) in a \(\mathbb Z^2\)-invariant way, and let \(\mathsf{K}\) be its associated Kasteleyn operator, that is, its weighted adjacency matrix. This operator commutes with the translation action of \(\mathbb Z^2\) and, in particular, it preserves the \(\mathbb Z^2\)-eigenspaces. These eigenspaces are indexed by characters of \(\mathbb Z^2\), that is, by a pair of Bloch-Floquet multipliers \((z, w)\in (\mathbb C^*)^2\). They are finite-dimensional with a distinguished basis \(\{\delta_v\}\) consisting of functions suppported on a single \(\mathbb Z^2\)-orbit and taking value 1 on a vertex \(v\) inside a fixed fundamental domain.

Let \(\mathsf{K}(z,w)\) be the matrix of \(\mathsf{K}\) in the basis \(\{\delta_v\}\) and acting on the \((z,w)\)-eigenspace of \(\mathbb Z^2\) and set, by definition, \(P(z,w)=\det \mathsf{K}(z,w)\). Different choices of the fundamental domain lead to polynomials that just differ by a factor of the form \(z^iw^j\). Thus the set \[ {\mathcal C}=\{(z,w)\in (\mathbb C^*)^2: P(z,w)=0\} \] is well defined and it is called the spectral curve associated to the original edge weights. The first result of the paper under review provides a new proof of the following (for a different proof see the paper by the authors and S. Sheffield [Ann. Math. (2), 163, No. 3, 1019–1056 (2006; Zbl 1154.82007)]): \({\mathcal C}\) is a degree \(d\) curve whose projective closure \(\overline{\mathcal C}\) is an \(M\)-curve, that is, the number of its connected components attains the maximum value \(1+(d-1)(d-2)/2\) among its degree. Moreover, \(\overline{\mathcal C}\) belongs to a distinguished class of \(M\)-curves, the so called Harnack curves, characterized by a very particular configuration of the connected components with respect to the three coordinate lines in \(\mathbb P^2(\mathbb R)\).

Notice that \(\overline{\mathcal C}\) intersects the coordinate lines of \(\mathbb P^2(\mathbb R)\) at \(3d\) points (counting multiplicities). This set of points is called the boundary of \(\overline{\mathcal C}\). Moreover, it has \(\binom {d-1}{2}\) compact components, and the authors prove that the divisor \((v)\) of any vertex \(v\) of \(\Gamma\) is a standard divisor on this curve, i.e. it is a \(\binom {d-1}{2}\)-degree divisor having precisely one point on each compact component of \(\overline{\mathcal C}\).

One of the main results of this fascinating and penetrating article is the bijectivity of the so called spectral transform, that is, the correspondence

\[ \begin{cases} \text{edge weights/gauge} \to \text{ Harnack curve}\\ \text{fundamental domain }\to \text{ ordering of boundary points}\\ \text{vertex }\to \text{ standard divisor.} \end{cases} \] The last two sections are devoted to understand the space of Harnack curves. Those having genus zero are studied in very detail. In particular, they are characterized as those Harnack curves that minimize the volume under their Ronkin function.

More in general the authors prove that Harnack curves with given boundary and genus \(g\) constitute a \(g\)-dimensional smooth semialgebraic variety with local coordinates given by the intercepts of the nontrivial compact ovals. Moreover, the neighbourhood of genus \(g\) and degree \(d\) Harnack curves inside the space of all degree \(d\) Harnack curves with the same boundary is diffeomorphic to \(\mathbb R^g\times [0, \infty)^ m\) where \(m={\binom {d-1}{2}-g}\) and the genus \(g\) stratum is embedded as \(\mathbb R^g\times \{0\}\).

This article and the already quoted preprint by the authors constitute a delightful piece of intriguing and deep mathematics which describe in a very elegant way unsuspected connections between different areas of mathematics.

Let \(\mathsf{K}(z,w)\) be the matrix of \(\mathsf{K}\) in the basis \(\{\delta_v\}\) and acting on the \((z,w)\)-eigenspace of \(\mathbb Z^2\) and set, by definition, \(P(z,w)=\det \mathsf{K}(z,w)\). Different choices of the fundamental domain lead to polynomials that just differ by a factor of the form \(z^iw^j\). Thus the set \[ {\mathcal C}=\{(z,w)\in (\mathbb C^*)^2: P(z,w)=0\} \] is well defined and it is called the spectral curve associated to the original edge weights. The first result of the paper under review provides a new proof of the following (for a different proof see the paper by the authors and S. Sheffield [Ann. Math. (2), 163, No. 3, 1019–1056 (2006; Zbl 1154.82007)]): \({\mathcal C}\) is a degree \(d\) curve whose projective closure \(\overline{\mathcal C}\) is an \(M\)-curve, that is, the number of its connected components attains the maximum value \(1+(d-1)(d-2)/2\) among its degree. Moreover, \(\overline{\mathcal C}\) belongs to a distinguished class of \(M\)-curves, the so called Harnack curves, characterized by a very particular configuration of the connected components with respect to the three coordinate lines in \(\mathbb P^2(\mathbb R)\).

Notice that \(\overline{\mathcal C}\) intersects the coordinate lines of \(\mathbb P^2(\mathbb R)\) at \(3d\) points (counting multiplicities). This set of points is called the boundary of \(\overline{\mathcal C}\). Moreover, it has \(\binom {d-1}{2}\) compact components, and the authors prove that the divisor \((v)\) of any vertex \(v\) of \(\Gamma\) is a standard divisor on this curve, i.e. it is a \(\binom {d-1}{2}\)-degree divisor having precisely one point on each compact component of \(\overline{\mathcal C}\).

One of the main results of this fascinating and penetrating article is the bijectivity of the so called spectral transform, that is, the correspondence

\[ \begin{cases} \text{edge weights/gauge} \to \text{ Harnack curve}\\ \text{fundamental domain }\to \text{ ordering of boundary points}\\ \text{vertex }\to \text{ standard divisor.} \end{cases} \] The last two sections are devoted to understand the space of Harnack curves. Those having genus zero are studied in very detail. In particular, they are characterized as those Harnack curves that minimize the volume under their Ronkin function.

More in general the authors prove that Harnack curves with given boundary and genus \(g\) constitute a \(g\)-dimensional smooth semialgebraic variety with local coordinates given by the intercepts of the nontrivial compact ovals. Moreover, the neighbourhood of genus \(g\) and degree \(d\) Harnack curves inside the space of all degree \(d\) Harnack curves with the same boundary is diffeomorphic to \(\mathbb R^g\times [0, \infty)^ m\) where \(m={\binom {d-1}{2}-g}\) and the genus \(g\) stratum is embedded as \(\mathbb R^g\times \{0\}\).

This article and the already quoted preprint by the authors constitute a delightful piece of intriguing and deep mathematics which describe in a very elegant way unsuspected connections between different areas of mathematics.

Reviewer: Jose Manuel Gamboa (Madrid)

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