Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

Details Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

2009-12-16

arxiv.org/abs/0912.3200v3

Mathematics

Combinatorics

Scientific paper

The aim of this paper is to formulate a discrete analog of the claim made by Alvarez-Gaume et al., (\cite{a}), stating that the partition function of the free fermion on a closed Riemann surface of genus $g$ is certain 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 $\D(s)(x)$, $s\in \S$, 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 $I(\D(s)(x))$ also called Feynman functions. By a result of Foata and Zeilberger $I(\D(s)(x))=\det(I-\D'(s)(x))$, where $\D'(s)(x)$ is obtained from $\D(s)(x)$ by replacing some entries by $0$. Each Feynman function is thus computable in a polynomial time. We suggest that in the case of critical embedding and bipartite graph $G$, the Feynman functions provide suitable discrete analogues of the Pfaffians of Dirac operators.

Affiliated with

Also associated with

No associations

Rating

Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-49473