Holographic timelike entanglement and $c$ theorem for supersymmetric QFTs in ($ 0+1 $)d

TL;DR

This work develops a holographic approach to timelike entanglement entropy (tEE) in ($0+1$)d SUSY QFTs, showing that tEE encodes the effective number of degrees of freedom along RG flows. By analyzing two explicit dual setups—$\mathcal{N}=2$ matrix models with massive deformations (NATD of $AdS_5 \times S^5$) and $\mathcal{N}=4$ SCQM quivers (AdS$_2$-based IIB backgrounds)—the authors demonstrate that the imaginary part of tEE tracks a flow central charge $c_{flow}$ and that tEE correlates with the holographic central charge ($c_{hol}$) and the quantum complexity $\mathcal{C}_V$, with UV poles of identical order ($3$) across observables. They show that UV renormalisation via disconnected counterterms yields finite tEE and that the UV behaviour of tEE matches the c-function predictions, supporting tEE as a robust DOF-counting measure for RG flows in ($0+1$)d. The results establish a close link between tEE, holographic c-functions, and complexity, suggesting a unified picture of DOF counting in low-dimensional SUSY QFTs and motivating extensions to other dimensions and models.

Abstract

We present a holographic set up that computes timelike Entanglement Entropy (tEE) in $ (0+1) $d QFTs preserving some amount of SUSY. The first example we consider is that of $\mathcal{N}=2$ matrix models with massive deformations. These are dual to non-Abelian T-dual of $AdS_5 \times S^5$ that asymptotes to \emph{smeared} D0 branes. The second example, that we consider is of $ \mathcal{N}=4 $ superconformal quantum mechanical quivers in ($ 0+1 $)d that are dual to a class of type IIB backgrounds with an $ AdS_2 $ factor. In both of these examples, tEE reveals a remarkable similarity with holographic $ c $ function pertaining to a RG flow. We further compute the complexity in these models, which also reveals an identical behaviour indicating the fact that tEE is a measure of number of degrees of freedom for these ($ 0+1 $)d SQFTs in a RG flow from UV to deep IR.

Figures (8)

• Figure 1: Extremal connected surface of type I with turning point at $z=z_0$.
• Figure 2: (a) Plot of imaginary tEE \ref{['e2.15']} with the location of the turning point ($z_0$). (b) Plot of subsystem size \ref{['e2.16']} with the location of the turning point ($z_0$).
• Figure 3: Plot of imaginary component \ref{['e2.15']} of tEE with the size \ref{['e2.16']} of the entangling region.
• Figure 4: A pair of connected surfaces without a turning point.
• Figure 5: A pair of disconnected surfaces that acts like a UV regulator.