Making the hyper--K\"ahler structure of N=2 quantum string manifest

TL;DR

This paper demonstrates that the Lorentz covariant formulation of N=2 string theory in curved space explicitly reveals a hyper--K"ahler structure, connecting superconformal currents to complex geometric structures.

Contribution

It shows how the hyper--K"ahler structure of N=2 string theory becomes manifest in a curved background through superconformal symmetry constraints.

Findings

Hyper--K"ahler structure is explicitly revealed in the covariant formulation.

Superconformal currents couple to a background two-form that is holomorphic and covariantly constant.

Three complex structures obeying a (pseudo)quaternion algebra are constructed.

Abstract

We show that the Lorentz covariant formulation of N=2 string in a curved space reveals an explicit hyper--K\"ahler structure. Apart from the metric, the superconformal currents couple to a background two--form. By superconformal symmetry the latter is constrained to be holomorphic and covariantly constant and allows one to construct three complex structures obeying a (pseudo)quaternion algebra.