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Kasteleyn operators from mirror symmetry

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David Treumann, Harold Williams, Eric Zaslow

Algebraic Geometry · Submitted 14 Oct 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same.

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arXiv ID1810.05985 (version 2)
Pith IDY7325IKF, Pith's permanent record of this paper and its review
CitationDavid Treumann, Eric Zaslow, Harold Williams. "Kasteleyn operators from mirror symmetry." arXiv:1810.05985 (2018). https://pith.science/paper/Y7325IKF
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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.30 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathbb{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathbb{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the asymptotic boundary of $L$.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. contribution 0.48 0.88 0.59 asserted 1.00
s1 The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathbb{C}^\times)^2$. method 0.59 0.74 0.15 asserted 0.96
s2 The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. method 0.53 0.83 0.25 asserted 0.97
s3 This lives on a stacky toric compactification of $(\mathbb{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). method 0.41 0.60 0.07 asserted 0.96
s4 We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. background 0.71 0.29 0.03 asserted 0.87
s5 We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the asymptotic boundary of $L$. result 0.89 0.94 0.09 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathbb{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathbb{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the asymptotic boundary of $L$.

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References
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3 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 2 sentences without a confident label

Cites it as background 1 paper
  • Monowalls relate to a number of significant problems involving non-abelian Hodge theory [ref] , mirror symmetry [ref] , Calabi-Yau moduli spaces and quantum gauge theories in five dimensions [ref] , and integrable systems [ref] .

    Crystal Volumes and Monopole Dynamics · Introduction · 0.95

What it cites · 32 sentences without a confident label

Builds on 1 paper
Compares with 1 paper
Cites as background 10 papers
  • We will work with the following categories of sheaves on $T^n$ [ref] . * $^c_Λ_(T^n)$, the dg category of constructible sheaves with asymptotic microsupport contained in $Λ_$, * $^w_Λ_(T^n)$, the dg category of wrapped constructible sheaves with asymptotic microsupport contained in $Λ_$.

    Wrapped microlocal sheaves on pairs of pants · The coherent-constructible correspondence · 0.93

  • Occasionally we cite basic results from [ref] which are stated for $^c_L(M)$ or $^c(M)$ but known to also hold for $_L(M)$ or $(M)$; we refer to [ref] or [ref] for a general discussion, citing [ref] without comment in the text.

    Brane structures in microlocal sheaf theory · Notation · 0.93

  • The Legendrian $Λ_$ can be identified with a skeleton of a generic fiber of the Hori-Vafa potential $W ∈[T_M]$ [ref] , where $T_M$ is the dual torus of $T_N ⊂_$ [ref] .

    Lagrangian Skeleta of Hypersurfaces in $(\mathbb{C}^*)^n$ · Further context · 0.87

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