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Symmetries of Holographic Super-Minimal Models

Kentaro Hanaki, Cheng Peng

TL;DR

This work derives the asymptotic symmetry of ${\cal N}=2$ higher-spin supergravity in AdS$_3$ and shows that it is given by the nonlinear super-$W_{\infty}[\lambda]$ algebra ${\mathcal S}{\cal W}_{\infty}[\lambda]$, with a wedge subalgebra isomorphic to the underlying higher-spin algebra ${\rm shs}[\lambda]$. By comparing to the large-$n$ limit of the ${\cal N}=2$ CP$^n$ Kazama–Suzuki model, the authors identify the boundary chiral algebra ${\cal S}{\cal W}_{n}$ with the bulk asymptotic symmetry, establishing two nontrivial checks: explicit matching of the lowest-spin commutators and identical spectra for degenerate representations of chiral primaries after appropriate limits. The results rely on a detailed Drinfeld–Sokolov reduction framework and a careful treatment of boundary conditions, gauge fixing, and mode expansions, all ensuring a consistent bulk–boundary dictionary. The findings strengthen the supersymmetric higher-spin AdS$_3$/CFT$_2$ duality and offer a concrete bridge to CP$^n$-type CFTs, suggesting further exploration of partition functions and higher-spin correlation functions as additional tests of the correspondence.

Abstract

We compute the asymptotic symmetry of the higher-spin supergravity theory in AdS_3 and obtain an infinite-dimensional non-linear superalgebra, which we call the super-W_infinity[lambda] algebra. According to the recently proposed supersymmetric duality between higher-spin supergravity in an AdS_3 background and the 't Hooft limit of the N=2 CP^n Kazama-Suzuki model on the boundary, this symmetry algebra should agree with the 't Hooft limit of the chiral algebra of the CFT, SW_n. We provide two nontrivial checks of the duality. By comparing the algebras, we explicitly match the lowest-spin commutation relations in the super-W_infinity[lambda] with the corresponding commutation relations in the 't Hooft limit on the CFT side. We also consider the degenerate representations of the two algebras and find that the spectra of the chiral primary fields are identical.

Symmetries of Holographic Super-Minimal Models

TL;DR

This work derives the asymptotic symmetry of higher-spin supergravity in AdS and shows that it is given by the nonlinear super- algebra , with a wedge subalgebra isomorphic to the underlying higher-spin algebra . By comparing to the large- limit of the CP Kazama–Suzuki model, the authors identify the boundary chiral algebra with the bulk asymptotic symmetry, establishing two nontrivial checks: explicit matching of the lowest-spin commutators and identical spectra for degenerate representations of chiral primaries after appropriate limits. The results rely on a detailed Drinfeld–Sokolov reduction framework and a careful treatment of boundary conditions, gauge fixing, and mode expansions, all ensuring a consistent bulk–boundary dictionary. The findings strengthen the supersymmetric higher-spin AdS/CFT duality and offer a concrete bridge to CP-type CFTs, suggesting further exploration of partition functions and higher-spin correlation functions as additional tests of the correspondence.

Abstract

We compute the asymptotic symmetry of the higher-spin supergravity theory in AdS_3 and obtain an infinite-dimensional non-linear superalgebra, which we call the super-W_infinity[lambda] algebra. According to the recently proposed supersymmetric duality between higher-spin supergravity in an AdS_3 background and the 't Hooft limit of the N=2 CP^n Kazama-Suzuki model on the boundary, this symmetry algebra should agree with the 't Hooft limit of the chiral algebra of the CFT, SW_n. We provide two nontrivial checks of the duality. By comparing the algebras, we explicitly match the lowest-spin commutation relations in the super-W_infinity[lambda] with the corresponding commutation relations in the 't Hooft limit on the CFT side. We also consider the degenerate representations of the two algebras and find that the spectra of the chiral primary fields are identical.

Paper Structure

This paper contains 15 sections, 110 equations.

Table of Contents

  1. Introduction
  2. Higher-spin supergravity as a Chern-Simons Theory
  3. Supersymmetric higher-spin algebra $shs[\lambda]$
  4. Boundary conditions, constraints and gauge fixing
  5. Global symmetry
  6. Super-$W_{\infty}[\lambda]$ algebra as the asymptotic symmetry
  7. Boundary condition for super-$W_{\infty}[\lambda]$ algebra
  8. Super-$W_{\infty}[\lambda]$ symmetries
  9. Identification with the ${\mathbb{C}}$P$^{n}$ chiral algebras
  10. Large $n$ limits of the ${\mathbb{C}}$P$^{n}$ chiral algebra
  11. Degenerate representations
  12. Conclusion
  13. Super-higher-spin algebra $shs[\lambda]$
  14. Nonlinear terms in super-$W_{\infty} [\lambda]$
  15. Useful formulae for $A(n,n-1)$ algebra