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We present a unified approach to holomorphic anomaly equations and somewell-known quantum spectral curves. We develop a formalism of abstract quantumfield theory based on the diagrammatics of the Deligne-Mumford moduli spaces$\overline{{\mathcal M}}_{g,n}$ and derive a quadratic recursion relation forthe abstract free energies in terms of the edge-cutting operators. Thisabstract quantum field theory can be realized by various choices of a sequenceof holomorphic functions or formal power series and suitable propagators, andthe realized quantum field theory can be represented by formal Gaussianintegrals. Various applications are given.

A Unified Approach to Holomorphic Anomaly Equations and Quantum Spectral Curves