{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:50Z","timestamp":1753893830840,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $X$.\u00a0 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.<\/jats:p>","DOI":"10.37236\/3741","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:33:27Z","timestamp":1578702807000},"source":"Crossref","is-referenced-by-count":0,"title":["Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions"],"prefix":"10.37236","volume":"22","author":[{"given":"Martin","family":"Loebl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Petr","family":"Somberg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2015,1,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p10\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:38:55Z","timestamp":1579257535000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/3741","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,1,9]]},"article-number":"P1.10"}}