[Paper Review] On Open/Closed String Duality

Justin Khoury, Herman Verlinde|ArXiv.org|Jan 10, 2000

Black Holes and Theoretical Physics3 references11 citations

TL;DR

This paper establishes that the open/closed string duality underlying the AdS/CFT correspondence manifests as a holographic renormalization group (RG) flow equation in both the large-N open string regime and the small ’t Hooft coupling limit. Using conformal equivalence between planar open string diagrams and closed string tree diagrams, the authors derive the same RG flow equation—previously obtained via supergravity in the strong coupling limit—microscopically from weakly coupled open string theory, confirming its universality across coupling regimes.

ABSTRACT

It was recently shown, using the AdS/CFT correspondence, that the low energy effective action of a large $N$ open string theory satisfies a holographic RG flow equation closely related to the Hamilton-Jacobi equation of 5-d supergravity. In this paper we re-obtain the same flow equation in the dual regime of small 't Hooft coupling $λ\ll 1$. Our derivation makes use of the conformal equivalence between planar open string diagrams and closed string tree diagrams. This equivalence can be viewed as a microscopic explanation of the open/closed string duality that underlies the AdS/CFT correspondence.

Motivation & Objective

• To extend the holographic RG flow equation—previously derived in the large ’t Hooft coupling regime via AdS/CFT—to the dual regime of small coupling.
• To provide a microscopic, worldsheet-based explanation for open/closed string duality by exploiting conformal equivalence between planar open and closed string diagrams.
• To demonstrate that the same RG flow structure (equation 3) holds in the weak coupling limit, with all quantities defined via weak-coupling expansions.
• To clarify the correspondence between worldsheet and spacetime RG flows, showing that UV/IR mixing is preserved via a scale-proportional mapping.

Proposed method

• Utilizes the conformal equivalence between planar multi-loop open string diagrams and closed string tree diagrams with D-brane boundary states.
• Represents all open string loop holes as external closed string states (D-brane boundary states) in the dual channel, converting UV divergences into IR divergences.
• Applies the recursive structure of counterterms from the Bogolyubov-Parasiuk-Hepp-Zimmerman (BPHZ) renormalization method to construct the effective action order-by-order.
• Derives the flow equation (20) by identifying the worldsheet beta-functions with the RG evolution of couplings, linking them to spacetime beta-functions via a scale-proportional factor.
• Uses the generating functional for the effective action $ S( heta) = S_E( heta) + ilde{ ho}( heta) $, with $ ilde{ ho} $ encoding non-local contributions from loop diagrams.
• Establishes the invariance of the total action $ S $ under the flow $ \dot{\phi}^i = \beta^i(\phi) $, showing that extremal solutions lie on complete RG trajectories.

Experimental results

Research questions

• RQ1 Can the holographic RG flow equation derived in the strong coupling regime via AdS/CFT be reproduced in the weak coupling limit using open string theory?
• RQ2 What is the microscopic worldsheet mechanism underlying open/closed string duality in the planar limit?
• RQ3 How does the conformal equivalence between open and closed string diagrams explain the universality of the RG flow structure?
• RQ4 What is the precise correspondence between worldsheet and spacetime RG flows in the context of string theory?
• RQ5 Does the RG invariance of the effective action persist across all values of the ’t Hooft coupling $ \lambda $, including $ \lambda \ll 1 $?

Key findings

• The same holographic RG flow equation (3) is reproduced in the weak coupling regime ($ \lambda \ll 1 $), confirming its universality across coupling scales.
• The conformal equivalence between planar open string diagrams and closed string tree diagrams provides a microscopic foundation for open/closed string duality.
• UV divergences in open string diagrams are mapped to IR divergences in the dual closed string channel, enabling a consistent RG interpretation.
• The effective action $ S(\phi) = S_E(\phi) + \Gamma(\phi) $ satisfies the same RG invariance condition as in the strong coupling case, with $ \Gamma(\phi) $ now defined via weak-coupling perturbation theory.
• The worldsheet beta-functions $ \beta^i_{\text{WS}} $ and spacetime beta-functions $ \beta^i_{\text{ST}} $ are related by a scale-proportional factor $ \gamma $, preserving the RG structure across scales.
• The recursive structure of the flow equation (25) closely mirrors the Bogolyubov recursion for QFT counterterms, suggesting a deep connection between string theory and renormalization in quantum field theory.