One-loop exactness of gravity partition function

In a previous work, we showed that the two- and three-loop contributions to the partition function of three-dimensional gravity in flat space vanish. This is in agreement with the expected one-loop exactness dictated by the underlying symmetry at the quantum level. To highlight the distinctive nature of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>D</a:mi> <a:mo>=</a:mo> <a:mn>3</a:mn> </a:math> case, we extend the three-loop computation to arbitrary spacetime dimensions <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>D</c:mi> </c:math> . In higher dimensions, the number of contributions increases substantially, reinforcing the view that one-loop exactness is a unique feature of three-dimensional gravity.