Spread complexity for the planar limit of holography

TL;DR

This paper extends the concept of spread complexity to supersymmetric systems, enabling analytical computation of complexity in the holographic planar limit by incorporating fermionic states and superalgebras.

Contribution

It introduces a generalized framework for spread complexity that includes fermionic and supercoherent states, applicable to semiclassical systems with Lie superalgebras.

Findings

Extended spread complexity to superalgebras.

Analyzed supercoherent states in super Heisenberg-Weyl and OSp(2|1).

Computed complexity for large charge superstring states.

Abstract

Complexity is a fundamental characteristic of states within a quantum system. Its use is however mostly limited to bosonic systems, inhibiting its present applicability to supersymmetric theories. This is also relevant to its application to the AdS/CFT correspondence. To address this limitation, we extend the framework of spread complexity beyond bosonic systems to include fermionic and supercoherent states. This offers a gateway to compute spread complexity analytically for any semiclassical system governed by a Hamiltonian associated with a Lie (super)algebra. This requires extending the Krylov chain to a Krylov path in a higher-dimensional lattice. A detailed analysis of supercoherent states within the super Heisenberg-Weyl and OSp$(2|1)$ algebras elucidates distinct contributions from bosonic and fermionic degrees of freedom to the complexity. This generalisation allows us to access the semiclassical regime of the planar limit of the holographic correspondence. We then compute the spread complexity of large charge superstring states on the gravity side, which are equivalent to the dual gauge states. The resulting complexity leads to Krylov paths capturing the geometry in which the string propagates.