Conformal field theory complexity from Euler-Arnold equations

Abstract

Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.

arXiv comment31 pages + appendicies, 2 figures, extended version of arXiv:2005.02415 v2: added references and minor improvements

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• Exact Gravity Duals for Simple Quantum Circuits
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• Fubini-Study geometries in the higher-dimensional gravity