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Emergent Strings from Quantum Field Theory

Guim Planella Planas

TL;DR

Addressing the lack of a generic derivation of holographic duals from quantum field theories, the paper develops a procedure that reexpresses perturbative QFT amplitudes as sums over two-dimensional geometries by using Schwinger parametrization and Strebel differential mappings of Feynman diagrams to Riemann surfaces. Restoring diffeomorphism and Weyl invariance promotes moduli-space integration to a path integral over worldsheet metrics, yielding a non‑critical string with a Liouville direction that acts as the holographic coordinate and aligning the string genus expansion with the QFT loop expansion at fixed genus $g$. Loop divergences are shown to correspond to standard string degeneration limits, implying gravitational backreaction is equivalent to QFT renormalization. The 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, with potential extensions to gauge theories and QCD-like holography.

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.

Emergent Strings from Quantum Field Theory

TL;DR

Addressing the lack of a generic derivation of holographic duals from quantum field theories, the paper develops a procedure that reexpresses perturbative QFT amplitudes as sums over two-dimensional geometries by using Schwinger parametrization and Strebel differential mappings of Feynman diagrams to Riemann surfaces. Restoring diffeomorphism and Weyl invariance promotes moduli-space integration to a path integral over worldsheet metrics, yielding a non‑critical string with a Liouville direction that acts as the holographic coordinate and aligning the string genus expansion with the QFT loop expansion at fixed genus . Loop divergences are shown to correspond to standard string degeneration limits, implying gravitational backreaction is equivalent to QFT renormalization. The 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, with potential extensions to gauge theories and QCD-like holography.

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.

Paper Structure

This paper contains 12 sections, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. From a diagram to a surface
  3. Strebel Differentials
  4. Riemann Surfaces from Feynman Diagrams
  5. The path integral over metrics
  6. Feynman rules from CFT correlation functions
  7. Embedding the QFT directions
  8. A non-geometric large-N matrix model
  9. Auxiliary fields
  10. Interactions, symmetry factors and external sources
  11. The worldsheet effective action
  12. Conclusions and outlook

Figures (2)

  • Figure 1: Graphical representation of the construction of a Riemann surface from a graph. In (a) we show the gluing of the strips around a loop in the $w$ plane. In (b) we show the gluing of the strips around a vertex in the $u$ plane. All the curves depicted correspond to real and imaginary lines on the $z$ plane. In (c) we present an example for a complete graph.
  • Figure 2: A loop of length $4$ can be split into loops of length $3$ in two distinct ways. For the dual graph (blue lines) this corresponds to resolving a vertex of order $4$ into two vertices of order $3$. In moduli space, they correspond to the two directions normal to a codimension $1$ hypersurface.