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Path-Integral Complexity for Perturbed CFTs

Arpan Bhattacharyya, Pawel Caputa, Sumit R. Das, Nilay Kundu, Masamichi Miyaji, Tadashi Takayanagi

TL;DR

This paper extends path-integral optimization to two-dimensional CFTs perturbed by relevant operators, showing how a metric and space-dependent couplings can be tuned to preserve the target quantum state while minimizing a normalization-based complexity functional. The authors derive a generalized complexity functional combining Liouville-type terms with perturbative corrections from the perturbation, and they connect the optimized geometry to a time slice of AdS with backreaction from the scalar. They provide explicit analyses in free scalar and fermion theories, develop a Wilsonian and local RG framework for running couplings, and obtain perturbative expressions for the metric corrections and complexity divergences, which are found to match AdS/CFT expectations in the holographic context. Overall, the work links holographic geometry emergence, RG flow, and quantum information measures, offering a concrete route to quantify complexity in perturbed CFTs and to compare with holographic notions of complexity.

Abstract

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.

Path-Integral Complexity for Perturbed CFTs

TL;DR

This paper extends path-integral optimization to two-dimensional CFTs perturbed by relevant operators, showing how a metric and space-dependent couplings can be tuned to preserve the target quantum state while minimizing a normalization-based complexity functional. The authors derive a generalized complexity functional combining Liouville-type terms with perturbative corrections from the perturbation, and they connect the optimized geometry to a time slice of AdS with backreaction from the scalar. They provide explicit analyses in free scalar and fermion theories, develop a Wilsonian and local RG framework for running couplings, and obtain perturbative expressions for the metric corrections and complexity divergences, which are found to match AdS/CFT expectations in the holographic context. Overall, the work links holographic geometry emergence, RG flow, and quantum information measures, offering a concrete route to quantify complexity in perturbed CFTs and to compare with holographic notions of complexity.

Abstract

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.

Paper Structure

This paper contains 21 sections, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General Formulation of Path-Integral Optimization
  3. Optimization Procedure
  4. Interpretation
  5. Path-integral Optimization of 2D CFTs and Relevant Peturbations
  6. Path-integral Optimization of 2D CFTs
  7. Path-integral Optimization of Perturbed 2D CFTs
  8. Free Field Examples
  9. Massive Free Scalar
  10. Massive Free Fermions
  11. Optimizing Relevant Perturbations in 2D CFTs
  12. Wilsonian RG Flow
  13. Local RG Flow
  14. Evaluation of the Normalization $N[e^{2\phi},\lambda_{\phi}]$
  15. Complexity Functional
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The sketch of the scale transformation and the correlation function. The left and right picture corresponds to the path-integral representation of correlation function before and after the optimization, respectively. A tensor is attached on the face of each of these cells. In the left picture there are tensor with 4 legs ($T_{a_1, a_2, a_3 a_4}$ ) attached on the faces of the cells. After the coarse graining procedure these tensors are replaced by some effective tensors ($T_{a_1, a_2, a_3, a_4, a_5}$) as shown in the right picture.