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Frenkel, Lepowsky, and Meurman constructed a vertex operator algebra (VOA)associated to any even, integral, Euclidean lattice. In the language ofphysics, these are examples of chiral conformal field theories. In this paper,we define non-chiral vertex operator algebra and some associated notions. Wethen give a construction of a non-chiral VOA associated to an even, integral,Lorentzian lattice and construct their irreducible modules. We obtain themoduli space of such modular invariant non-chiral VOAs based on self-dualLorentzian lattices of signature $(m,n)$ assuming the validity of a technicalresult about automorphisms of the lattice. We finally show that Narainconformal field theories in physics are examples of non-chiral VOA. Ourformalism helps us to identify the chiral algebra of Narain CFTs in terms of aparticular sublattice and break its partition function into sum of characters.

Non-Chiral Vertex Operator Algebra Associated To Lorentzian Lattices And Narain CFTs