Rationality of Lorentzian lattice CFTs and the associated modular tensor category

We discuss the construction and classification of the irreducible modules and intertwining operators between the irreducible modules of a rational Lorentzian lattice vertex operator algebra (LLVOA) based on an even, self-dual Lorentzian lattice Λ ⊂ Rm,n Λ ⊂ R m , n of signature ( m , n ). We also discuss various equivalent characterizations of rationality of the modular invariant LLCFT and recover (and slightly generalize) some old results of Wendland. Finally, we describe the standard construction of modular tensor category (MTC) associated with 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, realizes the D(m 8) D ( m mod 8 ) level 1 Kac–Moody MTC.