scholarly journals Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

2012 ◽  
pp. 1209-1244 ◽  
Author(s):  
David Cimasoni
Keyword(s):  
Riemann Surfaces ◽  
Dirac Operators ◽  
Discrete Dirac Operators

scholarly journals Sparse one-dimensional discrete Dirac operators II: Spectral properties

2011 ◽  
Vol 52 (7) ◽  
pp. 073501 ◽  
Author(s):  
S. L. Carvalho ◽  
C. R. de Oliveira ◽  
R. A. Prado
Keyword(s):  
Spectral Properties ◽  
Dirac Operators ◽  
One Dimensional ◽  
Discrete Dirac Operators

scholarly journals Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

10.37236/3741 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Martin Loebl ◽  
Petr Somberg
Keyword(s):  
Riemann Surface ◽  
Partition Function ◽  
Linear Combination ◽  
Dirac Operators ◽  
Finite Graph ◽  
Discrete Analogue ◽  
Square Roots ◽  
Closed Riemann Surface ◽  
Discrete Analogues ◽  
Discrete Dirac Operators

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., realizing the partition function of the free fermion on a closed Riemann surface of genus $g$ as a linear combination of $2^{2g}$ Pfaffians of Dirac operators. Let $G=(V,E)$ be a finite graph embedded in a closed Riemann surface $X$ of genus $g$, $x_e$ the collection of independent variables associated with each edge $e$ of $G$ (collected in one vector variable $x$) and $\S$ the set of all $2^{2g}$ spin-structures on $X$.  We introduce $2^{2g}$ rotations $rot_s$ and $(2|E|\times 2|E|)$ matrices $\Delta(s)(x)$, $s\in \Sigma$, of the transitions between the oriented edges of $G$ determined by rotations $rot_s$. We show that the generating function of the even sets of edges of $G$, i.e., the Ising partition function, is a linear combination of the square roots of $2^{2g}$ Ihara-Selberg functions $I(\Delta(s)(x))$ also called Feynman functions. By a result of Foata and Zeilberger $I(\Delta(s)(x))=\det(I-\Delta'(s)(x))$, where $\Delta'(s)(x)$ is obtained from $\Delta(s)(x)$ by replacing some entries by $0$. Thus each Feynman function is computable in a polynomial time. We suggest that in the case of critical embedding of a bipartite graph $G$, the Feynman functions provide suitable discrete analogues for the Pfaffians of Dirac operators.

Discrete Dirac Operators in Clifford Analysis

2007 ◽  
Vol 17 (3) ◽  
pp. 451-467 ◽  
Author(s):  
Nelson Faustino ◽  
Uwe Kähler ◽  
Frank Sommen
Keyword(s):  
Clifford Analysis ◽  
Dirac Operators ◽  
Discrete Dirac Operators

scholarly journals Location of Eigenvalues of Non-self-adjoint Discrete Dirac Operators

2020 ◽  
Vol 21 (7) ◽  
pp. 2193-2217 ◽  
Author(s):  
B. Cassano ◽  
O. O. Ibrogimov ◽  
D. Krejčiřík ◽  
F. Štampach
Keyword(s):  
Dirac Operators ◽  
Discrete Dirac Operators
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