Kleinian Geometry and the N=2 Superstring

TL;DR

This paper explores the geometric structures underlying the $N=2$ superstring, emphasizing its requirement for a four-dimensional self-dual spacetime and examining related twistor theory and spacetime solutions.

Contribution

It analyzes the geometry of the $N=2$ superstring, including self-dual spacetimes and twistor theory, providing insights into the theory's geometric foundations and solutions.

Findings

$N=2$ superstring requires a 4D self-dual spacetime

Real twistor theory is relevant to flat space geometry

Certain complex spacetimes satisfy the $eta$-function equations

Abstract

This paper is devoted to the exploration of some of the geometrical issues raised by the $N=2$ superstring. We begin by reviewing the reasons that $\beta$-functions for the $N=2$ superstring require it to live in a four-dimensional self-dual spacetime of signature $(--++)$, together with some of the arguments as to why the only degree of freedom in the theory is that described by the gravitational field. We then move on to describe at length the geometry of flat space, and how a real version of twistor theory is relevant to it. We then describe some of the more complicated spacetimes that satisfy the $\beta$-function equations. Finally we speculate on the deeper significance of some of these spacetimes.