Topological holography: The example of the D2-D4 brane system

Abstract

We propose a toy model for holographic duality. The model is constructed by embedding a stack of NN D2-branes and KK 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}_NGLN (resp. \mathrm{GL}_KGLK) gauge group. We propose that in the large NN limit the BF theory on \mathbb{R}^2ℝ2 is dual to the closed string theory on \mathbb{R}^2 \times \mathbb{R}_+ \times S^3ℝ2×ℝ+×S3 with the Chern-Simons defect on \mathbb{R} \times \mathbb{R}_+ \times S^2ℝ×ℝ+×S2. 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}_K𝔤𝔩K. 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 D3-D5 brane configuration in type IIB – using supersymmetric twist and \OmegaΩ-deformation.

Feynman diagrams and $Ω$-deformed M-theory

“…This section is prepared for a quick review of twisted holography for non-experts. The idea was introduced in [1] and studied in various examples [4,15,18,19,38,39] with or without Ω-deformation. The reader who is familiar with [4] can skip most of this section, 4 A similar line of development was made in [34,35], using twisted -cohomology, where is a particular combination of a supercharge Q and a conformal supercharge S [36].…”

“…In our case, since the closed strings completely decouple from the analysis, the back-reaction is only encoded in the interaction related to the open strings. More precisely, the back-reaction is already encoded in the 5d-1d interaction Lagrangian (38), a part of which we reproduce below.…”

Giant Wilson loops and AdS2/dCFT1

“…Yet another direction would be to understand the relation to the twisted holography discussed in [48] and make contact with gl(M ) Yangian discussed there. For this purpose, one needs to consider a product of M Wilson loops in the antisymmetric representations, and compute the correlators of insertions.…”

“…Another motivation comes from the relation to the so-called twisted holography [46][47][48][49][50]. The twisted holography refers to special examples of the AdS/CFT correspondence in which both the bulk and the boundary theories are topologically (or holomorphically) JHEP11(2020)064…”

“…Such theories are typically much simpler than the theories relevant for the fullfledged AdS/CFT correspondence, and therefore may provide a good starting point for understanding the duality in precise details. In [48], it was pointed out that there is one such example which involves the topological twist of the D2/D4 brane system: the boundary side is given by two-dimensional BF theory and a product of Wilson loops in the antisymmetric representations while the bulk side is the holomorphic Chern-Simons theory in four dimensions [51]. They further showed that the operator algebra living on the Wilson line is isomorphic to the Yangian.…”