A formalism of abstract quantum field theory of summation of fat graphs

Abstract: 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 $\mathcal F_g^\mu$, abstract free energy $\mathcal F_g$, abstract partition function $\mathcal Z$, and abstract $n$-point functions $\mathcal W_{g,n}$ 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 $\mathcal W_{g,n}$ 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 $Z_N^{\text{Herm}}$ which proves that $Z_N^{\text{Herm}}$ 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 Schr\"odinger type-equation which is equivalent to the quantum spectral curve. We conjecture that in general cases the realization of the quadratic recursion for $\mathcal W_{g,n}$ is the E-O recursion, where the spectral curve and Bergmann kernel are constructed from realizations of $\mathcal W_{0,1}$ and $\mathcal W_{0,2}$ respectively using the framework of emergent geometry.