# Topological Holography: The Example of The D2-D4 Brane System

### Submission summary

 As Contributors: Nafiz Ishtiaque · Seyed Faroogh Moosavian Arxiv Link: https://arxiv.org/abs/1809.00372v4 (pdf) Date accepted: 2020-07-14 Date submitted: 2020-07-02 02:00 Submitted by: Ishtiaque, Nafiz Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics Approach: Theoretical

### Abstract

We propose a toy model for holographic duality. The model is constructed by embedding a stack of $N$ D2-branes and $K$ D4-branes (with one dimensional intersection) in a 6D topological string theory. The world-volume theory on the D2-branes (resp. D4-branes) is 2D BF theory (resp. 4D Chern-Simons theory) with $\mathrm{GL}_N$ (resp. $\mathrm{GL}_K$) gauge group. We propose that in the large $N$ limit the BF theory on $\mathbb{R}^2$ is dual to the closed string theory on $\mathbb R^2 \times \mathbb R_+ \times S^3$ with the Chern-Simons defect on $\mathbb R \times \mathbb R_+ \times S^2$. As a check for the duality we compute the operator algebra in the BF theory, along the D2-D4 intersection -- the algebra is the Yangian of $\mathfrak{gl}_K$. We then compute the same algebra, in the guise of a scattering algebra, using Witten diagrams in the Chern-Simons theory. Our computations of the algebras are exact (valid at all loops). Finally, we propose a physical string theory construction of this duality using a D3-D5 brane configuration in type IIB -- using supersymmetric twist and $\Omega$-deformation.

### Ontology / Topics

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Published as SciPost Phys. 9, 017 (2020)

This is a resubmission with minor revision as per editorial recommendation.

### List of changes

1. Corrected several spellings, and a typo in eq. 5.
2. At the end of section 1 pointed out relevant new references that came out in the last couple of years.
3. Slightly expanded the introduction to appendix B to better clarify the motivation and logic behind the mathematical results to follow.
4. Some footnotes have been moved to the main text.
5. Commented on the special nature of the lack of backreaction in the 4d Chern-Simons theory after eq. 14 with references to literature with different examples with and without backreaction.

### Submission & Refereeing History

Resubmission 1809.00372v4 on 2 July 2020
Submission 1809.00372v3 on 7 April 2020

## Reports on this Submission

### Report

The authors have clarified several issues that were raised in the previous referee report, and have further improved the readability of the paper. Therefore, we are happy to recommend the paper for publication in its current form.

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