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

TL;DR

This paper extends Kac-Moody boundary conditions in AdS$_3$ gravity to include fermionic fields, revealing two supersymmetric structures, a richer asymptotic algebra, and a spectrum with only bosonic soft modes.

Contribution

It introduces two methods for fermionic extension of Kac-Moody boundary conditions in AdS$_3$ supergravity, linking to super-Virasoro algebra and exploring a novel algebraic and geometric structure.

Findings

Two fermionic extensions related to super-Virasoro algebra.

A richer asymptotic structure with constraints on fermionic potentials.

Spectrum contains only bosonic soft excitations.

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.