Complexity measures from geometric actions on Virasoro and Kac-Moody orbits

TL;DR

The paper analyzes a symmetry-restricted notion of computational complexity for 2d CFTs, showing that, with a suitably extended cost function, the complexity equals the on-shell geometric action on coadjoint orbits of Virasoro and Kac-Moody groups, including central-extension terms. It demonstrates exact equivalence between complexity and geometric actions for both Virasoro and KM symmetries, derives explicit optimal circuits, and uncovers gauge-invariance subtleties that necessitate boundary terms. By connecting to Liouville theory and AdS$_3$ gravity, the work also clarifies the relation between complexity and holographic/gravitational actions, and it introduces an Euler-Arnold-geometric route (KdV) to nontrivial, non-phase complexity. Overall, KM symmetries exhibit richer, nontrivial complexity structures than the Virasoro case, and the Euler-Arnold framework offers a promising avenue for nontrivial complexity definitions in conformal settings.

Abstract

We further advance the study of the notion of computational complexity for 2d CFTs based on a gate set built out of conformal symmetry transformations. Previously, it was shown that by choosing a suitable cost function, the resulting complexity functional is equivalent to geometric (group) actions on coadjoint orbits of the Virasoro group, up to a term that originates from the central extension. We show that this term can be recovered by modifying the cost function, making the equivalence exact. Moreover, we generalize our approach to Kac-Moody symmetry groups, finding again an exact equivalence between complexity functionals and geometric actions. We then determine the optimal circuits for these complexity measures and calculate the corresponding costs for several examples of optimal transformations. In the Virasoro case, we find that for all choices of reference state except for the vacuum state, the complexity only measures the cost associated to phase changes, while assigning zero cost to the non-phase changing part of the transformation. For Kac-Moody groups in contrast, there do exist non-trivial optimal transformations beyond phase changes that contribute to the complexity, yielding a finite gauge invariant result. Furthermore, we also show that the alternative complexity proposal of path integral optimization is equivalent to the Virasoro proposal studied here. Finally, we sketch a new proposal for a complexity definition for the Virasoro group that measures the cost associated to non-trivial transformations beyond phase changes. This proposal is based on a cost function given by a metric on the Lie group of conformal transformations. The minimization of the corresponding complexity functional is achieved using the Euler-Arnold method yielding the Korteweg-de Vries equation as equation of motion.