$Sp(8)$ invariant higher spin theory, twistors and geometric BRST formulation of unfolded field equations

Abstract

We discuss twistor-like interpretation of the $Sp(8)$ invariant formulation of 4d massless fields in ten dimensional Lagrangian Grassmannian $Sp(8)/P$ which is the generalized space-time in this framework. The correspondence space $\mathbf{C}$ is $SpH(8)/PH$ where $SpH(8)$ is the semidirect product of $Sp(8)$ with Heisenberg group $\HG$ and $PH$ is some quasiparabolic subgroup of $SpH(8)$. Spaces of functions on $Sp(8)/P$ and $SpH(8)/PH$ consist of $Q_P $ closed functions on $Sp(8)$ and $Q_{PH} $ closed functions on $SpH(8)$, where $Q_P $ and $Q_{PH}$ are canonical BRST operators of $P$ and $PH$. The space of functions on the generalized twistor space $\mathbf{T}$ identifies with the $SpH(8)$ Fock module. Although $\mathbf{T}$ cannot be realized as a homogeneous space, we find a nonstandard $SpH(8)$ invariant BRST operator $\QQ$ $(\QQ^2 =0)$ that gives rise to an appropriate class of functions via the condition $\QQ f=0$ equivalent to the unfolded higher--spin equations. The proposed construction is manifestly $Sp(8)$ invariant, globally defined and coordinate independent. Its Minkowski analogue gives a version of twistor theory with both types of chiral spinors treated on equal footing. The extensions to the higher rank case with several Heisenberg groups and to the complex case are considered. A relation with Riemann theta functions, that are $\QQ$-closed, is discussed.