# Quantum Null Energy Condition and its (non)saturation in 2d CFTs

### Submission summary

 As Contributors: Daniel Grumiller Arxiv Link: https://arxiv.org/abs/1901.04499v1 Date submitted: 2019-01-30 Submitted by: Grumiller, Daniel Submitted to: SciPost Physics Domain(s): Theoretical Subject area: High-Energy Physics - Theory

### Abstract

We consider the Quantum Null Energy Condition (QNEC) for holographic conformal field theories in two spacetime dimensions (CFT$_2$). We show that QNEC saturates for all states dual to vacuum solutions of AdS$_3$ Einstein gravity, including systems that are far from thermal equilibrium. If the Ryu-Takayanagi surface encounters bulk matter QNEC does not need to be saturated, whereby we give both analytical and numerical examples. In particular, for CFT$_2$ with a global quench dual to AdS$_3$-Vaidya geometries we find a curious half-saturation of QNEC for large entangling regions. We also address order one corrections from quantum backreactions of a scalar field in AdS$_3$ dual to a primary operator of dimension $h$ in a large central charge expansion and explicitly compute both, the backreacted Ryu--Takayanagi surface part and the bulk entanglement contribution to EE and QNEC. At leading order for small entangling regions the contribution from bulk EE exactly cancels the contribution from the back-reacted Ryu-Takayanagi surface, but at higher orders in the size of the region the contributions are almost equal while QNEC is not saturated. For a half-space entangling region we find that QNEC is gapped by $h/4$ in the large $h$ expansion.

###### Current status:
Has been resubmitted

### Ontology / Topics

See full Ontology or Topics database.

### Submission & Refereeing History

Resubmission 1901.04499v2 on 14 March 2019
Submission 1901.04499v1 on 30 January 2019

## Reports on this Submission

### Strengths

1) Simple and elegant proof of the saturation of the quantum null energy condition for 2d holographic CFTs corresponding to the vacuum solutions of AdS3 gravity.

2) Very detailed (analytical and numerical) analysis of various cases including far-from equilibrium setups, and cases with a bulk scalar and backreactions for which the quantum null energy condition is not saturated.

### Weaknesses

I don't quite see any definite weaknesses. Some parts of the paper present somewhat speculative statements, such as the concept of "Quantum equilibrium". So, in principle, I would like to see more supports for such statements, which make the manuscript more valuable. However, these type of statements appear only in the last section, and for the bulk of the paper, the authors are quite concrete. So, I see no problem.

### Report

In their paper, the authors studied the quantum null energy condition (QNEC) in 2d holographic CFTs. After providing a simple and elegant proof of the saturation of the QNEC for vacuum solutions of AdS3 Einstein gravity, they studied various cases, including far from equilibrium setups and cases with a bulk scalar field. In the latter case, the QNEC needs not to saturate. They also discussed quantum corrections. I found that the manuscript is interesting and sheds light on the QNEC, and hence recommend the paper to be published once the minor concerns in the following are properly addressed (see below).

### Requested changes

1) Throughout the paper, "QNEC" is sometimes used to refer actually the RHS of the QNEC in (1.1). For example, there are sentences like "QNEC diverges". These sentences are somewhat strange, as the QNEC per se cannot diverge. I suggest the authors to find a better way of phrasing these sentences.

2) I don't quite see the merit of the numerics presented in Sec. 3.2.
The approximate numerics solution was found by approximating the step function.
Just below the numerical paragraph, however the authors presented
the exact solution (3.6) and (3.7), without approximating the step function.

3) I don't quite understand what the author would mean by the operator interpretation.The LHS of (1.1) is a number (expectation value), and hence the RHS should also be a number. I also failed to see the connection between the twisted Sugawara construction and the identification (6.7). I would hope that the authors could elaborate on this. I guess that it's perhaps simply a matter of putting brackets < ... > in some expressions like (6.7 - 6.9)?

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: good

Author Daniel Grumiller on 2019-03-14
(in reply to Report 1 on 2019-03-04)
Category: