Moduli spaces of type II twists

Abstract

We provide a complete classification of twisting supercharges for type IIA supergravity in a flat background by determining the orbits of square-zero elements in the $\mathcal{N}=(1,1)$ super Poincaré algebra in ten dimensions under the action of the Spin group. Together with recent results for type IIB, this completes the description of the moduli space of twists for type II supergravities. We show that the type IIA nilpotence variety is the union of the eleven-dimensional nilpotence variety and a second component, confirming a conjecture of Eager, Saberi, and Walcher, and determine which twists of type IIA arise via dimensional reduction. Using the pure spinor formulation of the superstring, we construct maps from the moduli spaces of worldsheet twists to those of target space twists for both type IIA and type IIB, and find that these maps are not surjective. Precisely those twists of supergravity whose stabilizers are special holonomy groups---$G_2$ in type IIA, $\mathrm{Spin}(7)$ and $\mathrm{Sp}(4)$ in type IIB---have no corresponding background on the pure spinor worldsheet. Finally, we work out the action of T-duality relating twists of type IIA and type IIB, and find an obstruction to dualizing the $G_2$-twist of type IIA along certain directions.