Holographic Krylov complexity in confining gauge theories

A bstract We study a bulk holographic diagnostic of Krylov (spread) complexity in the Anabalón-Ross solitonic background, a top-down Type IIB solution describing a twisted-circle compactification of $$ \mathcal{N} $$ N = 4 SYM which flows to a gapped, confining, effectively three-dimensional theory in the infrared. Our calculation assumes the recently proposed relation between the rate of spread complexity and the proper radial momentum of an infalling massive probe; it should therefore be read as a controlled bulk test of this proposal, rather than as an independent boundary-field-theory derivation of Krylov complexity. We solve the radial geodesic problem analytically, in the supersymmetric case, in terms of elliptic functions, and verify the result through the UV and IR expansions. The finite end of the space makes the proper momentum finite and leads to bounded, oscillatory behaviour of the associated complexity proxy, in contrast with the unbounded pure-AdS result. We also explain how the oscillation time is controlled by the same geometric scale that sets the Kaluza-Klein and glueball scales. The non-supersymmetric case is treated in the appendix. The result gives a top-down arena in which to sharpen the proposed spread-momentum correspondence and to compare confinement-induced suppression of complexity growth with future direct field-theory calculations.