In this work, we formulate a path-integral optimization for two dimensional conformal field theories perturbed by relevant operators. We present several evidences how this optimization mechanism works, based on calculations in free field theories as well as general arguments of RG flows in field theories. Our optimization is performed by minimizing the path-integral complexity functional that depends on the metric and also on the relevant couplings. Then, we compute the optimal metric perturbatively and find that it agrees with the time slice of the hyperbolic metric perturbed by a scalar field in the AdS/CFT correspondence. Last but not the least, we estimate contributions to complexity from relevant perturbations.

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Published in Journal of High Energy Physics, v. 2018, issue 7, article 86, p. 1-25.

© The Author(s) 2018

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Funded by SCOAP3.

AB is supported by JSPS fellowship. AB and TT are supported by JSPS Grant-in-Aid for JSPS fellowship 17F17023. PC and TT are supported by the Simons Foundation through the "It from Qubit" collaboration. PC is supported by the JSPS starting grant KAKENHI 17H06787. NK and TT are supported by JSPS Grant-in-Aid for Scientific Research (A) No.16H02182. TT is also partially supported by World Premier International Research Center Initiative (WPI Initiative) from the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT). The work of SRD is partially supported by National Science Foundation grant NSF-PHY/1521045.