2021/08/24 by Zhiyuan Wang, Jian Zhou, Wang, Zhiyuan +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2108.10498
openalex publication_date 2021/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we present a formalism of abstract quantum field theory for fat graphs and its realizations. This is a generalization of an earlier work for stable graphs. We define the abstract correlators \mathcal Fgμ, abstract free energy \mathcal Fg, abstract partition function \mathcal Z, and abstract n-point functions \mathcal Wg,n to be formal summations of fat graphs, and derive quadratic recursions using edge-contraction/vertex-splitting operators, including the abstract Virasoro constraints, an abstract cut-and-join type representation for \mathcal Z, and a quadratic recursion for \mathcal Wg,n which resembles the Eynard-Orantin topological recursion. When considering the realization by the Hermitian one-matrix models, we obtain the Virasoro constraints, a cut-and-join representation for the partition function ZNHerm which proves that ZNHerm is a tau-function of KP hierarchy, a recursion for n-point functions which is known to be equivalent to the E-O recursion, and a Schrödinger type-equation which is equivalent to the quantum spectral curve. We conjecture that in general cases the realization of the quadratic recursion for \mathcal Wg,n is the E-O recursion, where the spectral curve and Bergmann kernel are constructed from realizations of \mathcal W0,1 and \mathcal W0,2 respectively using the framework of emergent geometry.