Pseudo-supersymmetry, Consistent Sphere Reduction and Killing Spinors for the Bosonic String

TL;DR

This paper introduces a concept of 'pseudo-supersymmetry' for the bosonic string, showing that consistent sphere reductions can be understood via a Killing spinor equation even without true supersymmetry.

Contribution

It demonstrates that a natural Killing spinor equation exists for the bosonic string, linking it to consistent sphere reductions through a generalized notion of supersymmetry.

Findings

Killing spinor equations exist for the bosonic string.

Projection of integrability conditions yields equations of motion.

Supports broader understanding of sphere reductions beyond supersymmetry.

Abstract

Certain supergravity theories admit a remarkable consistent dimensional reduction in which the internal space is a sphere. Examples include type IIB supergravity reduced on S^5, and eleven-dimensional supergravity reduced on S^4 or S^7. Consistency means that any solution of the dimensionally-reduced theory lifts to give a solution in the higher dimension. Although supersymmetry seems to play a role in the consistency of these reductions, it cannot be the whole story since consistent sphere reductions of non-supersymmetric theories are also known, such as the reduction of the effective action of the bosonic string in any dimension D on either a 3-sphere or a (D-3)-sphere, retaining the gauge bosons of SO(4) or SO(D-2) respectively. We show that although there is no supersymmetry, there is nevertheless a natural Killing spinor equation for the D-dimensional bosonic string. A projection of the full integrability condition for these Killing spinors gives rise to the bosonic equations of motion (just as happens in the supergravity examples). Thus it appears that by extending the notion of supersymmetry to "pseudo-supersymmetry" in this way, one may be able to obtain a broader understanding of a relation between Killing spinors and consistent sphere reductions.