• double-scaled SYK
• krylov complexity
• JT gravity
• holographic duality

The objective of this thesis is to compute a certain notion of complexity in the SYK model considered in the double-scaled limit. We will introduce the double-scaled limit and explain how a certain diagrammatic expansion in chord diagrams arises. We will then use the computational techniques allowed by this diagrammatic expansion to compute the Krylov complexity of the infinite temperature thermofield double. At this point, we can try to generalize the computation to operator Krylov complexity, which is the original contribution of this work. We find an analytical expression for the operator Krylov complexity in the double-scaled SYK, in a particular continuum approximation. Now we can ask ourselves if we can identify a gravitational dual for this quantity. It was previously known that SYK in the IR is dual to JT gravity. In this framework, it was shown that the dual to the Krylov complexity of the infinite-temperature thermofield double is the geodesic length of the wormhole. We argue that in the case of operator complexity, a sensible candidate for the duality is a geodesic length in a JT gravity theory after the insertion of a shockwave.