# Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 4, page 1209-1244
- ISSN: 1435-9855

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topCimasoni, David. "Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices." Journal of the European Mathematical Society 014.4 (2012): 1209-1244. <http://eudml.org/doc/277794>.

@article{Cimasoni2012,

abstract = {Let $\sum $ be a flat surface of genus $g$ with cone type singularities. Given a bipartite graph $\Gamma $ isoradially embedded in $\sum $, we define discrete analogs of the $2^\{2g\}$ Dirac operators on $\sum $. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair $\Gamma \subset \sum $ for these discrete Dirac operators to be Kasteleyn matrices of the graph $\Gamma $. As a consequence, if these conditions are met, the partition function of the dimer model on $\Gamma $ can be explicitly written as an alternating sum of the determinants of these $2^\{2g\}$ discrete Dirac operators.},

author = {Cimasoni, David},

journal = {Journal of the European Mathematical Society},

keywords = {perfect matching; dimer model; discrete complex analysis; isoradial graph; Dirac operator; Kasteleyn matrices; perfect matching; dimer model; discrete complex analysis; isoradial graph; Dirac operator; Kasteleyn matrices},

language = {eng},

number = {4},

pages = {1209-1244},

publisher = {European Mathematical Society Publishing House},

title = {Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices},

url = {http://eudml.org/doc/277794},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Cimasoni, David

TI - Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 4

SP - 1209

EP - 1244

AB - Let $\sum $ be a flat surface of genus $g$ with cone type singularities. Given a bipartite graph $\Gamma $ isoradially embedded in $\sum $, we define discrete analogs of the $2^{2g}$ Dirac operators on $\sum $. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair $\Gamma \subset \sum $ for these discrete Dirac operators to be Kasteleyn matrices of the graph $\Gamma $. As a consequence, if these conditions are met, the partition function of the dimer model on $\Gamma $ can be explicitly written as an alternating sum of the determinants of these $2^{2g}$ discrete Dirac operators.

LA - eng

KW - perfect matching; dimer model; discrete complex analysis; isoradial graph; Dirac operator; Kasteleyn matrices; perfect matching; dimer model; discrete complex analysis; isoradial graph; Dirac operator; Kasteleyn matrices

UR - http://eudml.org/doc/277794

ER -

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