Bulk reconstruction and gauge invariance

In this paper, we discuss the concept of bulk reconstruction, which involves mapping bulk operators into CFT operators to understand the emergence of spacetime and gravity. We argue that the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi><a:mo>=</a:mo><a:mi>∞</a:mi></a:math> approximation fails to capture crucial aspects of gravity, as it does not respect gauge invariance and lacks direct connections between energy and boundary metrics. Key concepts such as entanglement wedge reconstruction and holographic error correction codes, which are based on the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>N</c:mi><c:mo>=</c:mo><c:mi>∞</c:mi></c:math> theory, may be incorrect or require significant revision when finite <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>N</e:mi></e:math> effects are considered. We present explicit examples demonstrating discrepancies in bulk reconstructions and suggest that a gauge-invariant approach is necessary for an accurate understanding.