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Supersymmetric extensions of Kac-Moody boundary conditions in AdS$_3$ gravity

Nabamita Banerjee, Vedant Bhutra, Suvankar Dutta, Soumava Kundu

Abstract

We extend the Kac-Moody (KM) boundary conditions of AdS$_3$ gravity by incorporating fermionic fields. For $\mathcal{N}=(1,1)$ AdS$_3$ supergravity, we show that there are two possible ways to implement the fermionic extension. In the first, the extended KM boundary conditions are related to the standard super-Virasoro (VS) boundary conditions through a large gauge transformation realized by the super-Miura map between fields and chemical potentials, establishing a supersymmetric generalization of the KM-VS correspondence. In the second, a more general boundary configuration leads to strong constraints on the fermionic chemical potentials, yet offers a much richer asymptotic structure. It provides us a novel realization of the extended Kac-Moody algebra, and a geometric interpretation in terms of folds in the relativistic free-fermion droplet. Finally, we quantize the latter theory by promoting the classical Poisson brackets to (anti-)commutators, construct the corresponding Hilbert space, and show that the resulting spectrum contains only bosonic soft excitations, with no additional fermionic soft modes.

Supersymmetric extensions of Kac-Moody boundary conditions in AdS$_3$ gravity

Abstract

We extend the Kac-Moody (KM) boundary conditions of AdS gravity by incorporating fermionic fields. For AdS supergravity, we show that there are two possible ways to implement the fermionic extension. In the first, the extended KM boundary conditions are related to the standard super-Virasoro (VS) boundary conditions through a large gauge transformation realized by the super-Miura map between fields and chemical potentials, establishing a supersymmetric generalization of the KM-VS correspondence. In the second, a more general boundary configuration leads to strong constraints on the fermionic chemical potentials, yet offers a much richer asymptotic structure. It provides us a novel realization of the extended Kac-Moody algebra, and a geometric interpretation in terms of folds in the relativistic free-fermion droplet. Finally, we quantize the latter theory by promoting the classical Poisson brackets to (anti-)commutators, construct the corresponding Hilbert space, and show that the resulting spectrum contains only bosonic soft excitations, with no additional fermionic soft modes.
Paper Structure (23 sections, 148 equations, 1 figure)

This paper contains 23 sections, 148 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. A review of bosonic boundary conditions and their integrable structure
  3. 3d Einstein gravity as a Chern-Simons theory
  4. Kac-Moody boundary conditions
  5. Boundary Hamiltonians and Poisson structure
  6. Asymptotic symmetry algebra and conserved charges
  7. A relativistic free fermion droplet description
  8. Near horizon symmetries and soft hairy black holes
  9. Virasoro boundary conditions
  10. Relating the boundary conditions
  11. A minimal supersymmetric extension of Kac--Moody boundary conditions
  12. Motivating the choice of gauge connection
  13. Canonical generators and asymptotic symmetry algebra
  14. Boundary Hamiltonians and the super-KdV hierarchy
  15. A different fermionic extension of Kac--Moody boundary conditions
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Folds on Fermi surface dutta2023higher.