Odd dimensional nonlocal Liouville conformal field theories

We construct Euclidean Liouville conformal field theories in odd number of dimensions. The theories are nonlocal and non-unitary with a log-correlated Liouville field, a Q-curvature background, and an exponential Liouville-type potential. We study the classical and quantum properties of these theories including the finite entanglement entropy part of the sphere partition function F, the boundary conformal anomaly and vertex operators’ correlation functions. We derive the analogue of the even-dimensional DOZZ formula and its semi-classical approximation.