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 Schrodinger 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.
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