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Holographic RG Flow on the Defect and g-Theorem

Satoshi Yamaguchi

TL;DR

This work extends the holographic understanding of RG flows to defect CFTs by formulating a holographic g-function and proving a g-theorem. It analyzes a mass deformation of a defect via a D5-brane in the AdS$_5\times S^5$ background, deriving a Bogomolnyi-like flow equation and solving for a SUSY-preserving trajectory where the defect decouples in the IR. The authors then define the holographic g-function as $\ln g = -\alpha T_{rr}$ and prove its monotonic decrease along the flow under the weaker energy condition, confirming the expected defect-level decrease in degrees of freedom. These results provide a clear, calculable example of defect RG flows in holography and set the stage for studying more general defect-to-defect or defect-to-CFT flows and their stringy corrections.

Abstract

We investigate relevant deformation and the renormalization group flow in a defect conformal field theory from the point of view of the holography. We propose a candidate of g-function in the context of the holography, and prove the g-theorem: the g-function is monotonically non-increasing along the RG flow. We apply this g-theorem to the D5-brane solution which is an asymptotically AdS_4 x S^2 brane in AdS_5 x S^5. This solution corresponds to the mass deformation of the defect CFT. We checked that the g-function is monotonically non-increasing in this solution.

Holographic RG Flow on the Defect and g-Theorem

TL;DR

This work extends the holographic understanding of RG flows to defect CFTs by formulating a holographic g-function and proving a g-theorem. It analyzes a mass deformation of a defect via a D5-brane in the AdS background, deriving a Bogomolnyi-like flow equation and solving for a SUSY-preserving trajectory where the defect decouples in the IR. The authors then define the holographic g-function as and prove its monotonic decrease along the flow under the weaker energy condition, confirming the expected defect-level decrease in degrees of freedom. These results provide a clear, calculable example of defect RG flows in holography and set the stage for studying more general defect-to-defect or defect-to-CFT flows and their stringy corrections.

Abstract

We investigate relevant deformation and the renormalization group flow in a defect conformal field theory from the point of view of the holography. We propose a candidate of g-function in the context of the holography, and prove the g-theorem: the g-function is monotonically non-increasing along the RG flow. We apply this g-theorem to the D5-brane solution which is an asymptotically AdS_4 x S^2 brane in AdS_5 x S^5. This solution corresponds to the mass deformation of the defect CFT. We checked that the g-function is monotonically non-increasing in this solution.

Paper Structure

This paper contains 9 sections, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The solution representing the flow
  3. Setting
  4. Mass deformation and D-brane solution representing the flow
  5. Supersymmetry of the solution
  6. $g$-theorem
  7. $g$-function in higher dimensional defect CFT
  8. Holographic $g$-function
  9. Conclusion

Figures (3)

  • Figure 1: The form of the brane. There are no singularity on the brane at $\rho\to 0$.
  • Figure 2: The configuration of the D3-branes and the D5-brane. In the left picture, 3-5 string have massless mode and the defect preserves the scale invariance in the gauge theory. In the deformation we done here, the D5-brane and the D3-branes are separated, the 3-5 string become massive, and the scale invariance is violated.
  • Figure 3: Another picture of separating the D3-branes and the D5-brane. The surface represents the geometry around the D3-branes and the bold line represents the D5-brane. If we add the mass to the fields on the defect, the D5-brane vanishes in the halfway of the throat.