A Lax Operator for $d=2$ $N=2$ Supergravity

TL;DR

This paper derives a Lax operator for two-dimensional N=2 supergravity, demonstrating its integrability and extending previous results from higher supersymmetry cases.

Contribution

It provides the first explicit Lax operator formulation for N=2 supergravity in two dimensions, illustrating supergravity's integrability after dimensional reduction.

Findings

Derived the Lax operator for N=2 supergravity in 2D

Confirmed supergravity's integrability in reduced dimensions

Extended previous Lax operator results to lower supersymmetry

Abstract

General relativity and supergravity become integrable systems after dimensional reduction to two spacetime dimensions. This means the equations of motion can be encoded in the flatness condition for a Lax operator. Nicolai and Warner found Lax operators for dimensionally reduced supergravity. They gave explicit formulas primarily for the case with $N=16$ supersymmetry in two dimensions (which corresponds to $N=8$ supergravity in four dimensions). In this note, we derive analogous results for the case with $N=2$ supersymmetry in two dimensions (which corresponds to $N=1$ supergravity in four dimensions). This is the simplest example of the general fact that supergravity becomes an integrable system after dimensional reduction to two dimensions.