On a New Supersymmetric KdV Hierarchy in $2{-}d$ Quantum Supergravity

TL;DR

This paper demonstrates that a recently proposed supersymmetric extension of the KdV hierarchy in 2D quantum supergravity is fundamentally the same as the classical KdV hierarchy, with added supersymmetric features, and establishes its integrability.

Contribution

It proves the equivalence of the new supersymmetric KdV hierarchy to the classical one and extends it with supersymmetric odd flows, providing conserved charges and bihamiltonian structure.

Findings

Hierarchy is essentially the classical KdV with a superfield replacement.

Conserved charges and bihamiltonian structure are identified.

Hierarchy is extended with supersymmetric odd flows.

Abstract

Recently a new supersymmetric extension of the KdV hierarchy has appeared in a matrix-model-inspired approach to $2{-}d$ quantum supergravity. Here we prove that this hierarchy is essentially the KdV hierarchy, where the KdV field is now replaced by an even superfield. This allows us to find the conserved charges and the bihamiltonian structure, and to prove its integrability. We also extend the hierarchy by odd flows in a supersymmetric fashion.