Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

Cimasoni, 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 -