Extended affine Lie algebras, vertex algebras and equivariant ϕ -coordinated quasi-modules

For any nullity 2 extended affine Lie algebra ℰ of maximal type and ℓ ∈ ℂ, we prove that there exist a vertex algebra V ℰ ( ℓ ) and an automorphism group G of V ℰ ( ℓ ) equipped with a linear character χ , such that the category of restricted ℰ -modules of level ℓ is canonically isomorphic to the category of ( G, χ )-equivariant ϕ -coordinated quasi V ℰ ( ℓ )-modules. Moreover, when ℓ is a nonnegative integer, there is a quotient vertex algebra L ℰ ( ℓ ) of V ℰ ( ℓ ) modulo by a G -stable ideal, and we prove that the integrable restricted ℰ -modules of level ℓ are exactly the ( G, χ )-equivariant ϕ -coordinated quasi- L ℰ ( ℓ )-modules.