zbMATH — the first resource for mathematics

Planar dimers and Harnack curves. (English) Zbl 1100.14047
Let \(\Gamma\) 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.

14P15 Real-analytic and semi-analytic sets
14H50 Plane and space curves
Full Text: DOI arXiv
[1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Vol. I , Grundlehren Math. Wiss. 267 , Springer, New York, 1985. · Zbl 0559.14017
[2] V. I. Arnold and S. P. Novikov, eds., Dynamical Systems, VII , Encyclopaedia Math. Sci. 16 , Springer, Berlin, 1994.
[3] O. Babelon, D. Bernard, and M. Talon, Introduction to Classical Integrable Systems , Cambridge Monogr. Math. Phys., Cambridge Univ. Press, Cambridge, 2003. · Zbl 1045.37033
[4] A. Beauville, “Systèmes hamiltoniens complètement intégrables associés aux surfaces \(K3\)” in Problems in the Theory of Surfaces and Their Classification (Cortona, Italy, 1988) , Sympos. Math. 32 , Academic Press, London, 1991, 25–31. · Zbl 0827.58022
[5] -, Determinantal hypersurfaces , Michigan Math. J. 48 (2000), 39–64. · Zbl 1076.14534 · doi:10.1307/mmj/1030132707
[6] R. J. Cook and A. D. Thomas, Line bundles and homogeneous matrices , Quart. J. Math. Oxford Ser. (2) 30 (1979), 423–429. · Zbl 0437.14004 · doi:10.1093/qmath/30.4.423
[7] I. A. Dynnikov and S. P. Novikov, Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds , Russian Math. Surveys 52 (1997), 1057–1116. · Zbl 0928.35107 · doi:10.1070/RM1997v052n05ABEH002105
[8] -, Geometry of the triangle equation on two-manifolds , Mosc. Math. J. 3 (2003), 419–438. · Zbl 1046.39016
[9] D. Gieseker, H. KnöRrer, and E. Trubowitz, The Geometry of Algebraic Fermi Curves , Perspect. Math. 14 , Academic Press, Boston, 1993. · Zbl 0778.14011
[10] A. Gorsky, N. Nekrasov, and V. Rubtsov, Hilbert schemes, separated variables, and D-branes , Comm. Math. Phys. 222 (2001), 299–318. · Zbl 0985.81107 · doi:10.1007/s002200100503
[11] J. C. Hurtubise, Integrable systems and algebraic surfaces , Duke Math. J. 83 (1996), 19–50. · Zbl 0857.58024 · doi:10.1215/S0012-7094-96-08302-7
[12] P. Kasteleyn, Statistics of dimers on a lattice , Physica 27 (1961), 1209–1225. · Zbl 1244.82014
[13] R. Kenyon, The Laplacian and Dirac operators on critical planar graphs , Invent. Math. 150 (2002), 409–439. · Zbl 1038.58037 · doi:10.1007/s00222-002-0249-4
[14] -, An introduction to the dimer model , ICTP Lecture Notes Ser. 17 (2004), http://users.ictp.it/\(\sim\)pub_off/lectures/vol17.html R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and amoebae , to appear in Ann. of Math. (2), · Zbl 1154.82007 · doi:10.4007/annals.2006.163.1019 · arxiv.org
[15] G. Mikhalkin, Real algebraic curves, the moment map and amoebas , Ann. of Math. (2) 151 (2000), 309–326. JSTOR: · Zbl 1073.14555 · doi:10.2307/121119 · www.math.princeton.edu · eudml:125158
[16] G. Mikhalkin and H. Rullgå Rd, Amoebas of maximal area , Internat. Math. Res. Notices 2001 , no. 9, 441–451. · Zbl 0994.14032 · doi:10.1155/S107379280100023X
[17] A. A. Oblomkov, Difference operators on two-dimensional regular lattices , Theoret. and Math. Phys. 127 (2001), 435–445. · Zbl 0998.39012 · doi:10.1023/A:1010355707136
[18] J. Propp, Generalized domino-shuffling , Theoret. Comput. Sci. 303 (2003), 267–301. · Zbl 1052.68095 · doi:10.1016/S0304-3975(02)00815-0
[19] E. K. Sklyanin, “Separation of variables –.-new trends” in Quantum Field Theory, Integrable Models and Beyond (Kyoto, 1994) , Progr. Theoret. Phys. Suppl. 118 (1995), 35–60. · Zbl 0868.35002 · doi:10.1143/PTPS.118.35
[20] V. Vinnikov, Selfadjoint determinantal representations of real plane curves , Math. Ann. 296 (1993), 453–479. · Zbl 0789.14029 · doi:10.1007/BF01445115 · eudml:165095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.