On Toda equation and half BPS supergravity solution in M-theory

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.

1/4-BPS M-theory bubbles with SO(3) × SO(4) symmetry

“…It is expected that this particular Toda equation is responsible for the dynamics of 1/2-BPS operators of the superconformal field theory defined on M2 or M5-branes, although in this case we do not have a perturbative description of the dual conformal field theory and it is not clear how to derive the Toda system from the field theory. See [8,9] for discussions on the solutions of the Toda equation and their interpretations as giant gravitons.…”

Large N Vector Quantum Mechanics and Bubbling Supertube Solutions

“…where A is the monopole potential in the a space, which is given by 19) in the vicinity of a. This reproduces the Lorentz force term L 1 .…”

“…Bubbling solutions in M-theory are also studied in Refs. 17)- 19). Different droplets with the same asymptotic forms give different classical supergravity solutions with the same boundary conditions.…”