Quantum Complexity of Nonlocal Field Theories

TL;DR

The paper investigates holographic complexity in nonlocal field theories—specifically a dipole-deformed N=4 SYM and noncommutative (anisotropic and isotropic) SYM—using CV, subregion CV, and CA. It shows that nonlocality induces hyperscaling deviations and phase structure in subregion complexity that align with entanglement entropy transitions, while full-volume complexity often mirrors AdS-like behavior, sometimes due to dimensional coincidences. Action complexity reveals nontrivial UV-structure, including positivity bounds on nonlocal parameters and the necessity of topological terms in 10D, indicating limits to UV completeness in these theories. The results extend previous LST/WCFT findings and highlight rich, nonuniversal UV behavior of complexity in non-AdS holography, suggesting several future directions such as finite temperature/charge, subregion CA, and broader nonlocal backgrounds.

Abstract

Entanglement entropy for nonlocal field theories displays a universal ``volume law" scaling \cite{Barbon:2008ut, Karczmarek:2013xxa, Shiba:2013jja, Pang:2014tpa} as opposed to the ``area law" scaling for local field theories. The aim of this work is to determine whether complexity displays any such an universal scaling laws. The field theories considered here are obtained by deforming $\mathcal{N}=4$ SYM theory by higher dimension operators introducing nonlocality, namely a dipole deformation and noncommutativity (NCSYM) by turning on world volume Kalb-Ramond $B$ field. The dual gravity backgrounds have a running dilaton, in addition to the $B$-field background, which alter AdS asymptotics. Our results capture nonlocality in the hyperscaling behavior for complexity. We also compute the subregion complexity which display phase transitions in the nonlocal field theories with the transition point being the same as that for the phase transition of entanglement entropy \cite{Karczmarek:2013xxa}. These new results dovetail nicely with our findings from our previous works \cite{Chakraborty:2020fpt, Katoch:2022hdf, Bhattacharyya:2022ren} on other lower dimensional nonlocal field theories such as little string theories (LSTs) and warped conformal field theories (WCFTs).

Figures (3)

• Figure 1: Different numeric plots of $C_{V}$ Vs $l$ for five different values of $a=10^{-0.5},10^{-1.0},10^{-1.5},10^{-2.0},10^{-2.5}$ and $W=10^{15}$ and in the last we have shown the numeric plots of $\ln(C_{V})$ Vs $\ln(l)$. For all the plots we have considered AdS radius $R=1$ and $W=10^{15}$, and pointed out the co-ordinates for the transition point.
• Figure 2: Numeric plots of $\log(\mathcal{C}_V)$ vs $\log(l)$ for different values of $a=10^{-7}, 10^{-9}, 10^{-11}, 10^{-13}$ respectively. The transition points are indicated in respective plots.
• Figure 3: Plot of area vs subregion length for dipole deformed theory for $a=10^{-3.0}$ and $u_{b}=10^{2}$