Complexity from spinning primaries

We define circuits given by unitary representations of Lorentzian conformal field theory in 3 and 4 dimensions. Our circuits start from a spinning primary state, allowing us to generalize formulas for the circuit complexity obtained from circuits starting from scalar primary states. These results are nicely reproduced in terms of the geometry of coadjoint orbits of the conformal group. In contrast to the complexity geometry obtained from scalar primary states, the geometry is more complicated and the existence of conjugate points, signaling the saturation of complexity, remains open.

Automated summary

By defining circuits based on unitary representations of Lorentzian conformal field theory in 3 and 4 dimensions, we are able to generalize formulas for circuit complexity starting from spinning primary states. These results are effectively replicated through the geometry of coadjoint orbits of the conformal group. However, unlike the complexity geometry derived from scalar primary states, the geometry derived from spinning primary states is more intricate and the presence of conjugate points signaling complexity saturation is still unknown.