Abstract: Different proposals such as Complexity=Volume and the Python’s Lunch conjecture have linked complexity in the boundary theory to geometric features in the bulk within the AdS/CFT correspondence. In one concrete realization of this connection, the duality between JT gravity and the SYK model provides a mapping between the renormalized length of the wormhole in the bulk and the Krylov complexity in the boundary theory. Such a relation was obtained explicitly in the double-scaling limit of the SYK model. 

In this talk, I will present recent work on the Krylov complexity of the U(1) symmetric complex SYK model in the double-scaling limit. We show that, in the (n,Q)-basis of chord states, the grand canonical transfer matrix is block diagonal and thus the grand canonical Krylov complexity is just the sum over the canonical Krylov complexities in the charge sectors weighted by a probability function that depends on the grand potential.