Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm (1010.4036v2)

Abstract: We formulate a holographic Wilsonian renormalization group flow for strongly coupled systems with a gravity dual, motivated by the need to extract efficiently low energy behavior of such systems. Starting with field theories defined on a cut-off surface in a bulk spacetime, we propose that integrating out high energy modes in the field theory should correspond to integrating out a part of the bulk geometry. We describe how to carry out this procedure in practice in the classical gravity approximation using examples of scalar and vector fields. By integrating out bulk degrees of freedom all the way to a black hole horizon, this formulation defines a refined version of the black hole membrane paradigm. Furthermore, it also provides a derivation of the semi-holographic description of low energy physics.

• The paper introduces a framework that equates integrating short-distance bulk geometry with Wilsonian RG flow for holographic duality.
• It derives flow equations that transparently connect bulk field boundary conditions with effective field theory couplings.
• It reinterprets the membrane paradigm by modeling the stretched black hole horizon as capturing low-energy, hydrodynamic boundary dynamics.

Holographic Wilsonian RG and the Membrane Paradigm

The paper "Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm" by Thomas Faulkner, Hong Liu, and Mukund Rangamani offers an insightful paper into the implementation of the Wilsonian renormalization group (RG) flow within the framework of the gauge/gravity duality. With the increasing interest in the holographic interpretation of strongly coupled quantum field theories, this paper provides a crucial link by integrating the concept of Wilsonian renormalization into the holographic setup. The key focus is on how strong coupling phenomena in a boundary theory with a gravity dual can be understood through a bulk spacetime and RG-related processes.

Summary and Key Results

The authors propose a framework that equates integrating out short-distance degrees of freedom in a boundary field theory with integrating out regions of the bulk spacetime. This is motivated by the geometric nature of the AdS/CFT (Anti-de Sitter/Conformal Field Theory) correspondence, where the radial direction in the bulk correlates with the energy scale of the boundary theory. Starting with theories defined on a cut-off surface in the bulk gravity setup, the authors develop a practical procedure to integrate out geometric sections up to the black hole horizon.

Under this framework, they derive flow equations analogous to the Wilsonian RG for the boundary action $S_B$, effectively transparently connecting boundary conditions on the bulk fields with field theory couplings. The paper extends to practical applications involving scalar and vector fields, showcasing how this formalism aids in both conceptual understanding and computational methods.

One notable outcome is a refined formulation of the "membrane paradigm," offering a re-interpretation in the context of black hole horizons. The stretched horizon acts as an effective description for low-energy bulk dynamics, capturing boundary theory dynamics such as hydrodynamic diffusion. This demonstrates the potential for the extended Wilsonian approach to simplify the characterization of strongly coupled dynamics in holographic scenarios.

Implications and Future Directions

The implications of this research are significant for the paper of holographic dualities, particularly in non-trivial backgrounds like black holes. Providing a systematic approach to integrate out degrees of freedom and defining boundary actions, this work supports the exploration of semi-holographic descriptions that bridge the gap between fully holographic models and traditional effective field theory.

The incorporation of gauge fields and scalar fields into this flow picture suggests that this method can be expanded to cover a wide range of other fields and perhaps include gravitational degrees of freedom. Given this, future work might focus on developing fully-fledged effective actions involving gravitational perturbations and exploring their ability to decode complex, emergent phenomena in quantum gravity or strongly interacting systems.

Conclusion

This paper is a critical step toward understanding how holographic models can effectively incorporate ideas from RG theory. The development of the holographic Wilsonian RG flow opens avenues for interpreting low-energy boundary dynamics through well-posed geometric principles, enhancing our understanding of the underpinnings of the gauge/gravity duality.

Further advancements may lead to robust frameworks that can tackle both theoretical challenges in gravity and practical applications in quantum field theory, especially in the difficult but fascinating regime of strong coupling. Such work promises new insights into the deep connections tying together quantum theory, geometry, and emergent physical phenomena.