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.
Highlights
We consider the operator product expansion (OPE) algebra of gauge invariant local operators, we argue that this algebra can be computed in the bulk theory by computing a certain algebra of scatterings from the asymptotic boundary in the limit N ∞
Authors of [27] studied a twisted holography closely related to AdS3/CFT2 duality, they highlight in particular the link to Koszul duality and contains a rare introduction to Koszul duality. [28] computes certain operator algebra of a topological quantum mechanics living at the intersection of M2 and M5 branes in an Ωdeformed M-theory using Feynman diagram techniques similar to the ones we shall use in our computations
The operator algebra of the 2d BF theory consists of all theses operators but in this paper we focus on the non-commuting ones, in other words we, focus on the quotient of the full operator algebra of the boundary theory by its center
Summary
Holography is a duality between two theories, referred to as a bulk theory and a boundary theory, in two different space-time dimensions that differ by one [1,2,3]. In light of the connection between Koszul duality and holography, this result suggests that if there is a theory whose local operator algebra is the Yangian of glK that theory could be a holographic dual to the twisted 4d theory. The algebra of gauge invariant local operators along this D2-D4 intersection is precisely the Yangian of glK This connected the D2 world-volume theory and the D4 world-volume theory via holography in the sense of Koszul duality. [28] computes certain operator algebra of a topological quantum mechanics living at the intersection of M2 and M5 branes in an Ωdeformed M-theory using Feynman diagram techniques similar to the ones we shall use in our computations This M2-M5 brane setup in Ω-deformed M-theory is studied in [29, 30] in the context of twisted holography
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