Cone holography is a codimension-$n$ doubly holographic model, which can be
interpreted as the holographic dual of edge modes on defects. The initial model
of cone holography is based on mixed boundary conditions. This paper formulates
cone holography with Neumann boundary conditions, where the brane-localized
gauge fields play an essential role. Firstly, we illustrate the main ideas in
an $text{AdS}_4/text{CFT}_1$ toy model. We show that the $U(1)$ gauge field
on the end-of-the-world brane can make the typical solution consistent with
Neumann boundary conditions. Then, we generalize the discussions to general
codimension-$n$ cone holography by employing brane-localized $p$-form gauge
fields. We also investigate perturbative solutions and prove the mass spectrum
of Kaluza-Klein gravitons is non-negative. Furthermore, we prove that cone
holography obeys holographic $c$-theorem. Finally, inspired by the recently
proposed chiral model in AdS/BCFT, we construct another type of cone holography
with Neumann boundary conditions by applying massive vector (Proca) fields on
the end-of-the-world brane.
Cone holography is a model that can be interpreted as the holographic dual of edge modes on defects. In this paper, we explore cone holography with Neumann boundary conditions, focusing on the role of brane-localized gauge fields. We start by discussing an $text{AdS}_4/text{CFT}_1$ toy model, where we show how the $U(1)$ gauge field on the end-of-the-world brane can produce solutions consistent with Neumann boundary conditions.
We then extend our discussions to general codimension-$n$ cone holography, utilizing brane-localized $p$-form gauge fields. We also study perturbative solutions and demonstrate that the mass spectrum of Kaluza-Klein gravitons is non-negative, providing important insights into the dynamics of cone holography.
Additionally, we establish that cone holography satisfies the holographic $c$-theorem, which is a crucial property for holographic models. This result further reinforces the validity and usefulness of cone holography.
Lastly, we introduce another variation of cone holography with Neumann boundary conditions. This new model incorporates massive vector (Proca) fields on the end-of-the-world brane, inspired by the recently proposed chiral model in AdS/BCFT.
Future Roadmap:
1. Further Investigation of $text{AdS}_4/text{CFT}_1$ Toy Model
- Explore different gauge fields on the end-of-the-world brane
- Investigate the effects of varying parameters in the model
- Study the behavior of other boundary conditions
2. Generalization to Higher Codimension Cone Holography
- Extend the study to codimension-$n$ cone holography
- Analyze the impact of different brane-localized $p$-form gauge fields
- Investigate the impact of higher-dimensional defects
3. Further Examination of Perturbative Solutions
- Explore additional perturbative solutions in cone holography
- Investigate the stability and physical properties of these solutions
- Analyze the effects of higher-order perturbations
4. Validation of Holographic $c$-Theorem
- Investigate the holographic $c$-theorem further
- Explore its implications in different holographic models
- Analyze the connection between cone holography and other models satisfying the $c$-theorem
5. Study of the New Variation of Cone Holography with Proca Fields
- Investigate the dynamics and properties of cone holography with massive vector (Proca) fields
- Explore possible applications and connections to chiral models
- Analyze the behavior under different boundary conditions and brane setups
While these future directions hold great promise, there are also potential challenges that need to be addressed. These include:
- The need for more advanced mathematical techniques to analyze higher codimension cone holography and perturbative solutions
- The computational complexity involved in exploring diverse parameter spaces in the different models
- Theoretical and experimental validation of the holographic $c$-theorem in various scenarios
- The need for a deeper understanding of the physical implications and applications of the new variation of cone holography with Proca fields
Overall, by addressing these challenges and exploring the potential opportunities, the future of cone holography looks promising. It holds the potential to provide further insights into holography, defects, and their connections to other areas of theoretical physics.