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Holographic Krylov complexity in ${\cal N}=4$ SYM

Ali Fatemiabhari, Horatiu Nastase, Dibakar Roychowdhury

TL;DR

The paper develops a holographic Krylov complexity for ${\cal N}=4$ SYM by modeling motion in ${\rm AdS}_5$ sliced by ${\rm AdS}_3$, identifying the ${\rm Sl}(2)$ subsector with motion in the AdS3 subspace. It extends a known relation between the time derivative of Krylov complexity and the proper momentum, $\dot C(t) = -P_{\bar{\rho}}/\epsilon$, to the AdS5 context via a proper-distance coordinate ${\bar{\rho}}$ with $ds^2 = d\bar{\rho}^2$. The analysis first treats the exact AdS3 subspace dual to the ${\rm Sl}(2)$ sector and then generalizes to ${\rm AdS}_5$ sliced by ${\rm AdS}_3$, providing explicit expressions for the canonical momenta and the proper momentum in global and Poincaré coordinates. The results illuminate how holographic Krylov complexity encodes the time evolution of the ${\rm Sl}(2)$ sector and offer a framework for extending to deformed or non-conformal settings, with future work including direct Krylov-complexity calculations in spin-chain realizations and further AdS slicings.

Abstract

We propose and calculate a holographic Krylov complexity in ${\cal N}=4$ SYM via the proper momentum for motion in $AdS_5$ sliced by $AdS_3$. The motion in an $AdS_3$ subgroup corresponds to the Krylov complexity of the $Sl(2)$ subsector. The general motion corresponds to the Krylov complexity of the ${\cal N}=4$ SYM.

Holographic Krylov complexity in ${\cal N}=4$ SYM

TL;DR

The paper develops a holographic Krylov complexity for SYM by modeling motion in sliced by , identifying the subsector with motion in the AdS3 subspace. It extends a known relation between the time derivative of Krylov complexity and the proper momentum, , to the AdS5 context via a proper-distance coordinate with . The analysis first treats the exact AdS3 subspace dual to the sector and then generalizes to sliced by , providing explicit expressions for the canonical momenta and the proper momentum in global and Poincaré coordinates. The results illuminate how holographic Krylov complexity encodes the time evolution of the sector and offer a framework for extending to deformed or non-conformal settings, with future work including direct Krylov-complexity calculations in spin-chain realizations and further AdS slicings.

Abstract

We propose and calculate a holographic Krylov complexity in SYM via the proper momentum for motion in sliced by . The motion in an subgroup corresponds to the Krylov complexity of the subsector. The general motion corresponds to the Krylov complexity of the SYM.

Paper Structure

This paper contains 14 sections, 77 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Krylov complexity and holographic proposal
  3. $AdS_3$ subspace vs. $Sl(2)$ sector
  4. $AdS_3$ subspace of $AdS_5\times S^5$ and its Krylov complexity
  5. $Sl(2)$ subsector of ${\cal N}=4$ SYM and its Krylov complexity
  6. Motion in $AdS_5$ sliced by $AdS_3$ and holographic Krylov complexity
  7. Motion in $AdS_5$ sliced by $AdS_3$
  8. Global coordinates for $AdS_3$
  9. Poincaré coordinates for $AdS_3$
  10. Proper momentum as derivative of Krylov complexity in ${\cal N}=4$ SYM
  11. Global coordinates for $AdS_3$
  12. Poincaré coordinates for $AdS_3$
  13. Field theory interpretation
  14. Conclusions

Figures (4)

  • Figure 1: The evolution of $r(t)$ for $AdS_3$ in global coordinates that follows from \ref{['dotr2']}. The initial conditions are set as $\dot r_0=0.1$, $\rho_0=4, r_0=3$, together with $t_0=0$.
  • Figure 2: We evaluate $r(t)$ for $AdS_3$ in Poincaré coordinates following \ref{['e4.39']}. The initial conditions are set to be $\dot r_0=0.01$, $\rho_0=8, r_0=7$.
  • Figure 3: The evolution of the proper momentum $P_{\bar{\rho}}(t)$ for $AdS_3$ in global coordinates has been carried out following \ref{['e4.44']}. The initial conditions are set to be $\dot r_0=0.1$, $\rho_0=4, r_0=3$, $t_0=0$.
  • Figure 4: The proper momentum for $AdS_3$ in Poincaré coordinates has been estimated using \ref{['e4.49']}. The initial conditions are set to be $\dot r_0=0.01$, $\rho_0=8, r_0=7$.