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 F µ g , abstract free energy Fg, abstract partition function Z, and abstract n-point functions 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 Z, and a quadratic recursion for 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 Z Herm N which proves that Z Herm N 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 Wg,n is the E-O recursion, where the spectral curve and Bergmann kernel are constructed from realizations of W 0,1 and W 0,2 respectively using the framework of emergent geometry. Contents1. Introduction 1.1. Backgrounds 1.2. The formalism of abstract quantum field theories and their realizations 1.3. Description of main results 1.4. Plan of the paper 2. Abstract Quantum Field Theory for Fat Graphs 2.1. Graphs on oriented surfaces 2.2.