Holography and Koszul duality: the example of the M2M2 brane

May 6, 2017
110 pages
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Abstract: (arXiv)
Si Li and author suggested in that, in some cases, the AdS/CFT correspondence can be formulated in terms of the algebraic operation of Koszul duality. In this paper this suggestion is checked explicitly for M2M2 branes in an Ω\Omega-background. The algebra of supersymmetric operators on a stack of KK M2M2 branes is shown to be Koszul dual, in large KK, to the algebra of supersymmetric operators of 1111-dimensional supergravity in an Ω\Omega-background (using the formulation of supergravity in an Ω\Omega-background presented in arXiv:1610.04144). The twisted form of supergravity that is used here can be quantized to all orders in perturbation theory. We find that the Koszul duality result holds to all orders in perturbation theory, in both the gravitational theory and the theory on the M2M2. (However, there is a certain non-linear identification of the coupling constants on each side which I was unable to determine explicitly). It is also shown that the algebra of operators on KK M2M2 branes, as KK \to \infty, is a quantum double-loop algebra (a two-variable analog of the Yangian). This algebra is also the Koszul dual of the algebra of operators on the gravitational theory. An explicit presentation for this algebra is presented, and it is shown that this algebra is the unique quantization of its classical limit. Some conjectural applications to enumerative geometry of Calabi-Yau threefolds are also presented.
  • operator: algebra
  • supersymmetry: algebra
  • algebra: duality
  • approximation: classical
  • field theory: conformal
  • membrane model
  • supergravity
  • perturbation theory
  • quantization
  • gravitation
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