Abstract
We initiate quantitative studies of complexity in (1+1)-dimensional conformal field theories with a view that they provide the simplest setting to find a gravity dual to complexity. Our work pursues a geometric understanding of complexity of conformal transformations and embeds Fubini-Study state complexity and direct counting of stress tensor insertion in the relevant circuits in a unified mathematical language. In the former case, we iteratively solve the emerging integro-differential equation for sample optimal circuits and discuss the sectional curvature of the underlying geometry. In the latter case, we recognize that optimal circuits are governed by Euler-Arnold type equations and discuss relevant results for three well-known equations of this type in the context of complexity.
Highlights
One of the most perplexing recent results in quantum gravity are holographic complexity proposals
We addressed the problem of complexity of unitary operators resulting from exponentiation of the right- moving component of the stress-energy tensor in CFT1+1, see (1) and (2)
There are three important features of the complexity notions we consider: Firstly, they lead to equations of motion second order in derivatives of τ, which allows to search for optimal circuits connecting two elements of the Virasoro group
Summary
One of the most perplexing recent results in quantum gravity are holographic complexity proposals. Drawing a parallel from the field of entanglement entropy, which includes the matching of the results of [21,22] by the holographic entanglement entropy [23], CFTs1+1 should be viewed as an ideal setting for accelerating our understanding of complexity in QFTs and holography. While we intend to present our main results in a concise and self-contained manner, some further discussions can be found in our companion work [30]
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