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

TL;DR

This paper introduces a toy holographic duality model based on D-brane configurations in topological string theory, connecting 2D BF theory with 4D Chern-Simons theory and computing their operator algebras exactly.

Contribution

It constructs a novel holographic duality model involving D2 and D4 branes in topological string theory and verifies it through exact algebra computations.

Findings

The operator algebra in BF theory is the Yangian of gl_K.

The algebra computed via Witten diagrams matches the BF theory result.

The duality is supported by an explicit string theory construction using D3-D5 branes.

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.