On certain global conformal invariants and 3-surface twistors of initial data sets

The Chern–Simons functionals built from various connections determined by the initial data hμν , χμν on a 3-manifold Σ are investigated. First it is shown that for asymptotically flat data sets the logarithmic fall off for hμν and rχμν is the necessary and sufficient condition of the existence of these functionals. The functional Y(k,l), built in the vector bundle corresponding to the irreducible representation of SL(2,C) labelled by (k, l), is shown to be determined by the Ashtekar–Chern–Simons functional and its complex conjugate. Y(k,l) is conformally invariant precisely in the l = k (i.e. tensor) representations. An unexpected connection with twistor theory is found: Y(k,k) can be written as the Chern–Simons functional built from the 3-surface twistor connection, and the not identically vanishing spinor parts of the 3-surface twistor curvature are given by the variational derivatives of Y(k,k) with respect to hμν and χμν . The time derivative Ẏ(k,k) of Y(k,k) is another global conformal invariant of the initial data set, and for vanishing Ẏ(k,k), in particular for all Petrov III. and N spacetimes, the Chern–Simons functional is a conformal invariant of the whole spacetime.