Rationality of Lorentzian Lattice CFTs And The Associated Modular Tensor Category

We discuss the rationality of Lorentzian lattice conformal field theory (LLCFT) recently constructed in arXiv:2312.16296 and obtain equivalent characterizations of rationality generalising Wendland's rational Narain CFT characterization. We then describe the construction of a modular tensor category (MTC) associated to rational LLCFTs. We explicitly construct the modular data and braiding and fusing matrices for the MTC. As a concrete example, we show that the LLCFT based on a certain even, self-dual Lorentzian lattice of signature $(m,n)$ with $m$ even realises the $D(m\bmod 8)$ level 1 Kac-Moody MTC.