Emergent Strings from Quantum Field Theory

TL;DR

This paper presents a method to reformulate certain quantum field theories as emergent string theories by mapping Feynman diagrams to Riemann surfaces and constructing a corresponding worldsheet conformal field theory, revealing a holographic duality.

Contribution

It introduces a novel geometric approach to derive string theories directly from quantum field theories using Strebel differentials and moduli space integration.

Findings

Feynman diagrams are mapped to Riemann surfaces via Strebel differentials.

Constructed a worldsheet conformal field theory encoding QFT Feynman rules.

Showed loop divergences correspond to string degeneration limits.

Abstract

We develop a procedure that reorganizes the perturbative expansion in a class of quantum field theories into a stringy amplitude expressed as a sum over two-dimensional geometries. Using Schwinger parametrization and the one-to-one correspondence between metric ribboned graphs and the moduli space of Riemann surfaces established via Strebel differentials, we map each Feynman diagram to a surface. We then construct a conformal field theory on the worldsheet whose correlation functions encode the full set of QFT Feynman rules directly from the geometry of the associated Riemann surface. Restoring diffeomorphism and Weyl invariance promotes the integral over moduli space to a path integral over worldsheet metrics, yielding a non-critical string theory whose Liouville mode naturally becomes a holographic direction. By construction, the expansion of the string theory amplitudes in the number of boundary state insertions matches the loop expansion in the QFT at fixed genus. Moreover, loop divergences are shown to match standard string-theoretic degeneration limits, indicating that gravitational backreaction is equivalent to QFT renormalization. Our construction provides a microscopic route from generic QFTs to emergent string theories and offers a framework for deriving holographic duals directly from field-theoretic data.