Topological holography: The example of the D2-D4 brane system
Nafiz Ishtiaque, Seyed Faroogh Moosavian, Yehao Zhou
Abstract
We propose a toy model for holographic duality. The model is constructed by embedding a stack of N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> D2-branes and K <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>K</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="normal"> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> </mml:mstyle> <mml:mi>N</mml:mi> </mml:msub> </mml:math> (resp. \mathrm{GL}_K <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="normal"> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> </mml:mstyle> <mml:mi>K</mml:mi> </mml:msub> </mml:math> ) gauge group. We propose that in the large N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> limit the BF theory on \mathbb{R}^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msup> </mml:math> is dual to the closed string theory on \mathbb{R}^2 \times \mathbb{R}_+ \times S^3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> with the Chern-Simons defect on \mathbb{R} \times \mathbb{R}_+ \times S^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mo>×</mml:mo> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔤</mml:mi> <mml:mi>𝔩</mml:mi> </mml:mstyle> <mml:mi>K</mml:mi> </mml:msub> </mml:math> . 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 \Omega <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Ω</mml:mi> </mml:math> -deformation.