Holographic complexity and non-commutative gauge theory

TL;DR

The late time holographic complexity growth rate, based on the “complexity equals action” conjecture, experiences an enhancement when the non-commutativity is turned on and this enhancement saturates a new limit which is exactly 1/4 larger than the commutative value.

Abstract

We study the holographic complexity of noncommutative field theories. The four-dimensional $\mathcal{N}=4$ noncommutative super Yang-Mills theory with Moyal algebra along two of the spatial directions has a well known holographic dual as a type IIB supergravity theory with a stack of D3 branes and non-trivial NS-NS B fields. We start from this example and find that the late time holographic complexity growth rate, based on the "complexity equals action" conjecture, experiences an enhancement when the non-commutativity is turned on. This enhancement saturates a new limit which is exactly 1/4 larger than the commutative value. We then attempt to give a quantum mechanics explanation of the enhancement. Finite time behavior of the complexity growth rate is also studied. Inspired by the non-trivial result, we move on to more general setup in string theory where we have a stack of D$p$ branes and also turn on the B field. Multiple noncommutative directions are considered in higher $p$ cases.

Figures (7)

• Figure 1: Two WDW patches separated by $\delta t$. Although the boundary of each patch is really at some large but finite $r_b$, the choice of $r_b$ drops out in the differences we consider and we do not indicate it explicitly in this graphic.
• Figure 2: Late time action growth rate normalized by $C=\frac{\alpha^4 \Omega_5 V_3}{\hat{g}_s^2}$ and extra $r_H$ dependence, versus $a r_H$, which is the Moyal scale measured in units of thermal length. It is observed that the complexification rate under the CA conjecture increases significantly when the Moyal scale is comparable to the thermal scale, and saturate a new bound which is 5/4 of the commutative value when the Moyal scale is much larger than the thermal scale.
• Figure 3: This circuit represents the end of one copy of a circuit $Q_U$ implementing a hypothetical unitary $U$ and the beginning of a second copy of $Q_U$. In this plot horizontal lines are qubits, and the dots connected by vertical lines are gates acting on the pair of qubits they connect. For this illustration, we will consider gates to be their own inverse. Gates from two copies may cancel (illustrated here with dashed blue lines connecting the gates), reducing the complexity of the circuit and providing a more efficient way to compute $U^N$. This cancellation relies, however, on the ability of gates to commute past each other, so that gates which could cancel can meet. We argue that in the noncommutative case, fewer gates commute and so there are fewer cancelations of this type. In this illustration, we see on the third line that a gate which can commute to cancel in the commutative case is prevented from doing so in the non-commutative case due to mild non-locality. Cartoon inspired by one used in a talk by Adam Brown.
• Figure 4: Normalized complexification rate versus time in thermal units for $\gamma = 80$ and $b=0$.
• Figure 5: normalized complexification rate versus time in thermal units. $\gamma$ is held fixed at 80 while $b=ar_H$ is varied.