Complexity from Spinning Primaries

TL;DR

This paper explores the complexity of circuits in Lorentzian conformal field theories starting from spinning primary states, generalizing previous scalar-based formulas and relating results to the geometry of coadjoint orbits.

Contribution

It introduces a new framework for calculating circuit complexity from spinning primaries and connects these results to the geometry of conformal group coadjoint orbits.

Findings

Complexity formulas are extended to spinning primaries.

The geometry of complexity is more intricate for spinning primaries.

Existence of conjugate points and complexity saturation remains unresolved.

Abstract

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.