On Toda Equation and Half BPS Supergravity Solution in M-Theory

TL;DR

This paper investigates scale-invariant solutions of the three-dimensional Toda equation relevant to half BPS supergravity in M-theory, revealing algebraic structures and explicit solutions including the PP-wave case.

Contribution

It demonstrates that scale-invariant solutions of the Toda equation satisfy a cubic algebraic equation, providing new explicit solutions and insights into supergravity configurations.

Findings

Scale-invariant solutions obey a cubic algebraic equation.

Explicit solutions include the PP-wave supergravity solution.

Provides a new algebraic perspective on supergravity solutions.

Abstract

Recently, it was shown that half BPS Supergravity solution of theories with SU(2$|$4) symmetry algebra is given uniformly by determining a single function which obeys three dimensional continuous Toda equation. In this paper, we study the scale invariant solution of Toda equation. Our motivation is that some solutions of half BPS sector of IIB supergravity, as one excepts from the fermion description of the theory, are scale invariant. By defining two auxiliary functions we prove that such solutions of Toda equation obey cubic algebraic equation. We obtain some simpl solutions of Toda equation specially, we observe that the PP-wave solution can be written in this fashion.