Hopf Algebraic Structure for Tagged Graphs and Topological Recursion
Abstract
Using the shuffle structure of the graphs, we introduce a new kind of the Hopf algebraic structure for tagged graphs with, or without loops. Like a quantum group structure, its product is non-commutative. With the help of the Hopf algebraic structure, after taking account symmetry of the tagged graphs, we reconstruct the topological recursion on spectral curves proposed by B. Eynard and N. Orantin, which includes the one-loop equations of various matrix integrals as special cases.
Cite
@article{arxiv.1607.08136,
title = {Hopf Algebraic Structure for Tagged Graphs and Topological Recursion},
author = {Xiang-Mao Ding and Yuping Li and Lingxian Meng},
journal= {arXiv preprint arXiv:1607.08136},
year = {2017}
}
Comments
Substantially improved version: We discussed on tagged graphs instead of unlabelled graphs; the definition of coproduct is changed; Section 5 is rewritten. One of main results on the relationships between coproduct and topological recursion is more specific and essential; title and abstract changed accordingly; v1 26 pages; v2 28pages