Toroidal extended affine Lie algebras and vertex algebras

In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $phi$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $phi$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.